A276588 Square array A(row,col) = Sum_{k=0..row} binomial(row,k)*(1+col+k)!, read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

1, 2, 3, 6, 8, 11, 24, 30, 38, 49, 120, 144, 174, 212, 261, 720, 840, 984, 1158, 1370, 1631, 5040, 5760, 6600, 7584, 8742, 10112, 11743, 40320, 45360, 51120, 57720, 65304, 74046, 84158, 95901, 362880, 403200, 448560, 499680, 557400, 622704, 696750, 780908, 876809, 3628800, 3991680, 4394880, 4843440, 5343120, 5900520, 6523224, 7219974, 8000882, 8877691

Offset

0,2

Links

Antti Karttunen, Table of n, a(n) for n = 0..1274; the first 50 antidiagonals of array

Index entries for sequences related to factorial base representation

Index entries for triangles and arrays related to Pascal's triangle

Formula

A(row,col) = Sum_{k=0..row} binomial(row,k)*A000142(1+col+k).

A(row,col) = A276075(A066117(row+1,col+1)).

Example

The top left corner of the array:
     1,     2,     6,     24,     120,      720,      5040,      40320
     3,     8,    30,    144,     840,     5760,     45360,     403200
    11,    38,   174,    984,    6600,    51120,    448560,    4394880
    49,   212,  1158,   7584,   57720,   499680,   4843440,   51932160
   261,  1370,  8742,  65304,  557400,  5343120,  56775600,  661933440
  1631, 10112, 74046, 622704, 5900520, 62118720, 718709040, 9059339520

Mathematica

T[r_, c_]:=Sum[Binomial[r, k](1 + c + k)!, {k, 0, r}]; Table[T[c, r - c], {r, 0, 10}, {c, 0, r}] // Flatten (* Indranil Ghosh, Apr 11 2017 *)

Prog

(Scheme)
(define (A276588 n) (A276588bi (A002262 n) (A025581 n)))
(define (A276588bi row col) (A276075 (A066117bi (+ 1 row) (+ 1 col)))) ;; Code for A066117bi given in A066117, and for A276075 under the respective entry.
(PARI) T(r, c) = sum(k=0, r, binomial(r, k)*(1 + c + k)!);
for(r=0, 10, for(c=0, r, print1(T(c, r - c), ", "); ); print(); ) \\ Indranil Ghosh, Apr 11 2017
(Python)
from sympy import binomial, factorial
def T(r, c): return sum([binomial(r, k) * factorial(1 + c + k) for k in range(r + 1)])
for r in range(11): print([T(c, r - c) for c in range(r + 1)]) # Indranil Ghosh, Apr 11 2017

Crossrefs

Transpose: A276589.

Topmost row (row 0): A000142, Row 1: A001048 (without its initial 2), Row 2: A001344 (from a(1) = 11 onward), Row 3: A001345 (from a(1) = 49 onward), Row 4: A001346 (from a(1) = 261 onward), Row 5: A001347 (from a(1) = 1631 onward).

Leftmost column (column 0): A001339, Column 1: A001340, Columns 2-3: A001341 & A001342 (apparently).

Cf. A276075.

Cf. also arrays A066117, A276586, A099884, A255483.

Sequence in context: A242340 A033766 A129340 * A275951 A276586 A280094

Adjacent sequences: A276585 A276586 A276587 * A276589 A276590 A276591

Keyword

nonn,tabl

Author

Antti Karttunen, Sep 19 2016

Status

approved