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A276588
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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), ...
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8
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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
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OFFSET
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0,2
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LINKS
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FORMULA
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A(row,col) = Sum_{k=0..row} binomial(row,k)*A000142(1+col+k).
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(Scheme)
(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
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CROSSREFS
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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).
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KEYWORD
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AUTHOR
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STATUS
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approved
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