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URL: https://oeis.org/A276588

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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), ...
8
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
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: A033766 A129340 A393338 * A275951 A276586 A280094
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, Sep 19 2016
STATUS
approved