Problem:
Score : 100 points

Problem Statement

There are N children, numbered 1, 2, \ldots, N.

They have decided to share K candies among themselves.
Here, for each i (1 \leq i \leq N), Child i must receive between 0 and a_i candies (inclusive).
Also, no candies should be left over.

Find the number of ways for them to share candies, modulo 10^9 + 7.
Here, two ways are said to be different when there exists a child who receives a different number of candies.

Constraints

All values in input are integers.

1 \leq N \leq 100

0 \leq K \leq 10^5

0 \leq a_i \leq K

Input

Input is given from Standard Input in the following format:

N K
a_1 a_2 \ldots a_N

Output

Print the number of ways for the children to share candies, modulo 10^9 + 7.

Sample Input 1

3 4
1 2 3

Sample Output 1

5

There are five ways for the children to share candies, as follows:

(0, 1, 3)

(0, 2, 2)

(1, 0, 3)

(1, 1, 2)

(1, 2, 1)

Here, in each sequence, the i-th element represents the number of candies that Child i receives.

Sample Input 2

1 10
9

Sample Output 2

0

There may be no ways for the children to share candies.

Sample Input 3

2 0
0 0

Sample Output 3

1

There is one way for the children to share candies, as follows:

(0, 0)

Sample Input 4

4 100000
100000 100000 100000 100000

Sample Output 4

665683269

Be sure to print the answer modulo 10^9 + 7.

Solution in Common Lisp (SBCL 1.1.14) (judge verdict: Accepted)

#-(or child-sbcl swank)
(quit
 :unix-status
 (process-exit-code
  (run-program *runtime-pathname*
               '("--control-stack-size" "32MB"
                 "--noinform" "--disable-ldb" "--lose-on-corruption" "--end-runtime-options"
                 "--eval" "(push :child-sbcl *features*)"
                 "--script" #.(namestring *load-pathname*))
               :output t :error t :input t)))

(eval-when (:compile-toplevel :load-toplevel :execute)
  (defparameter OPT
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0)))
  #+swank (progn (ql:quickload '(:cl-debug-print :fiveam))
                 (shadow :run)
                 (use-package :fiveam)))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)

;; BEGIN_INSERTED_CONTENTS

(defmacro nlet (name args &body body)
  (labels ((ensure-list (x) (if (listp x) x (list x))))
    (let ((args (mapcar #'ensure-list args)))
      `(labels ((,name ,(mapcar #'car args) ,@body))
         (,name ,@(mapcar #'cadr args))))))

;; (with-memoizing (:hash-table :test #'equal :key #'cons)
;;   (defun ...))
;; (with-memoizing (:array (10 10 * 10) :initial-element -1 :element-type 'fixnum)
;;   (defun ...))
(defmacro with-memoizing (cache-attribs def-form)
  (let* ((cache-attribs (if (atom cache-attribs) (list cache-attribs) cache-attribs))
         (cache-type (first cache-attribs))
         (dimensions-with-* (when (eql cache-type :array) (second cache-attribs)))
         (dimensions (remove '* dimensions-with-*))
         (rank (length dimensions))
         (rest-attribs (ecase cache-type
                         (:hash-table (cdr cache-attribs))
                         (:array (cddr cache-attribs))))
         (key (prog1 (getf rest-attribs :key) (remf rest-attribs :key)))
         (cache-form (case cache-type
                       (:hash-table `(make-hash-table ,@rest-attribs))
                       (:array `(make-array ',dimensions ,@rest-attribs))))
         (initial-element (when (eql cache-type :array)
                            (assert (member :initial-element rest-attribs))
                            (getf rest-attribs :initial-element))))
    (let ((value (gensym))
	  (present-p (gensym))
          (name-alias (gensym))
	  (args-lst (gensym))
          (indices (loop repeat rank collect (gensym))))
      (labels ((make-cache-check-form (cache-type args)
                 (case cache-type
                   (:hash-table
                    `(let ((,args-lst (funcall ,(or key #'list) ,@args)))
                       (multiple-value-bind (,value ,present-p)
                           (gethash ,args-lst cache)
                         (if ,present-p
                             ,value
                             (setf (gethash ,args-lst cache)
                                   (,name-alias ,@args))))))
                   (:array
                    (let ((memoized-args (loop for dimension in dimensions-with-*
                                               for arg in args
                                               unless (eql dimension '*)
                                               collect arg)))
                      (if key
                          `(multiple-value-bind ,indices
                               (funcall ,key ,@memoized-args)
                             (let ((,value (aref cache ,@indices)))
                               (if (eql ,initial-element ,value)
                                   (setf (aref cache ,@indices)
                                         (,name-alias ,@args))
                                   ,value)))
                          `(let ((,value (aref cache ,@memoized-args)))
                             (if (eql ,initial-element ,value)
                                 (setf (aref cache ,@memoized-args)
                                       (,name-alias ,@args))
                                 ,value)))))))
               (make-reset-form (cache-type)
                 (case cache-type
                   (:hash-table `(setf cache (make-hash-table ,@rest-attribs)))
                   (:array `(prog1 nil
                              (fill (array-storage-vector cache) ,initial-element)))))
               (make-reset-name (name)
                 (intern (format nil "RESET-~A" (symbol-name name))))
               (extract-declarations (body)
                 (remove-if-not (lambda (form) (eql 'declare (car form))) body)))
        (ecase (car def-form)
          ((defun)
           (destructuring-bind (_ name args &body body) def-form
             (declare (ignore _))
             `(let ((cache ,cache-form))
                (defun ,(make-reset-name name) () ,(make-reset-form cache-type))
                (defun ,name ,args
                  (labels ((,name-alias ,args ,@body))
                    (declare (inline ,name-alias))
                    ,@(extract-declarations body)
                    ,(make-cache-check-form cache-type args))))))
          ((nlet)
           (destructuring-bind (_ name bindings &body body) def-form
             (declare (ignore _))
             `(let ((cache ,cache-form))
                (nlet ,name ,bindings
                      ,(let ((args (mapcar (lambda (x) (if (atom x) x (car x))) bindings)))
                         `(labels ((,name-alias ,args ,@body))
                            (declare (inline ,name-alias))
                            ,@(extract-declarations body)
                            ,(make-cache-check-form cache-type args)))))))
          ((labels flet)
           (destructuring-bind (_ definitions &body labels-body) def-form
             (declare (ignore _))
             (destructuring-bind (name args &body body) (car definitions)
               `(let ((cache ,cache-form))
                  (,(car def-form)
                   ((,(make-reset-name name) () ,(make-reset-form cache-type))
                    (,name ,args
                           (labels ((,name-alias ,args ,@body))
                             (declare (inline ,name-alias))
                             ,@(extract-declarations body)
                             ,(make-cache-check-form cache-type args)))
                    ,@(cdr definitions))
                   (declare (ignorable #',(make-reset-name name)))
                   ,@labels-body))))))))))

;; (test with-memoizing
;;   (finishes (macroexpand `(with-memoizing (:hash-table :test #'equal)
;;                             (defun add (x y) (+ x y)))))
;;   (finishes (macroexpand `(with-memoizing (:array '(10 10)
;;                                            :element-type 'fixnum
;;                                            :initial-element -1)
;;                             (defun add (x y) (+ x y)))))
;;   (finishes (macroexpand `(with-memoizing (:array '(10 10)
;;                                            :element-type 'fixnum
;;                                            :initial-element -1)
;;                             (labels ((add (x y) (+ x y))
;; 		                     (my-print (x) (print x)))
;; 	                      (add 1 2))))))

(declaim (inline split-ints-into-vector))
(defun split-ints-into-vector (string dest-vector &key (offset 0))
  (declare (string string)
           ((simple-array * (*)) dest-vector)
           ((integer 0 #.most-positive-fixnum) offset))
  (loop for idx from offset below (length dest-vector)
        for pos1 = 0 then (1+ pos2)
        for pos2 = (position #\space string :start pos1 :test #'char=)
        do (setf (aref dest-vector idx)
                 (parse-integer string :start pos1 :end pos2))
        finally (return dest-vector)))

(defmacro define-int-types (&rest bits)
  `(progn
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "UINT~A" b)) () '(unsigned-byte ,b))) bits)
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "INT~A" b)) () '(signed-byte ,b))) bits)))
(define-int-types 2 4 7 8 15 16 31 32 62 63 64)

(defmacro println (obj &optional (stream '*standard-output*))
  `(let ((*read-default-float-format* 'double-float))
     (prog1 (princ ,obj ,stream) (terpri ,stream))))

;; Hauptteil

(defconstant +magic+ (+ 7 (expt 10 9)))
(defun solve (as k)
  (declare #.OPT
           ((simple-array uint32 (*)) as))
  (let ((n (length as)))
    (with-memoizing (:array (100 100001) :element-type 'uint32 :initial-element #xffffffff)
      (nlet recurse ((x (- n 1)) (y k))
        (cond ((zerop x)
               (if (<= y (aref as 0))
                   1
                   0))
              ((zerop y) 1)
              (t (mod (the fixnum (+ (recurse (- x 1) y)
                                     (recurse x (- y 1))
                                     (if (< (aref as x) y)
                                         (- (recurse (- x 1) (- y (aref as x) 1)))
                                         0)))
                      +magic+)))))))

(defun main ()
  (let* ((n (read))
         (k (read))
         (as (make-array n :element-type 'uint32)))
    (split-ints-into-vector (read-line) as)
    (println (solve as k))))

#-swank(main)