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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), ...

%I A276588 #27 Apr 09 2021 12:05:44
%S A276588 1,2,3,6,8,11,24,30,38,49,120,144,174,212,261,720,840,984,1158,1370,
%T A276588 1631,5040,5760,6600,7584,8742,10112,11743,40320,45360,51120,57720,
%U A276588 65304,74046,84158,95901,362880,403200,448560,499680,557400,622704,696750,780908,876809,3628800,3991680,4394880,4843440,5343120,5900520,6523224,7219974,8000882,8877691
%N 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), ...
%H A276588 Antti Karttunen, <a href="/A276588/b276588.txt">Table of n, a(n) for n = 0..1274; the first 50 antidiagonals of array</a>
%H A276588 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%H A276588 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A276588 A(row,col) = Sum_{k=0..row} binomial(row,k)*A000142(1+col+k).
%F A276588 A(row,col) = A276075(A066117(row+1,col+1)).
%e A276588 The top left corner of the array:
%e A276588      1,     2,     6,     24,     120,      720,      5040,      40320
%e A276588      3,     8,    30,    144,     840,     5760,     45360,     403200
%e A276588     11,    38,   174,    984,    6600,    51120,    448560,    4394880
%e A276588     49,   212,  1158,   7584,   57720,   499680,   4843440,   51932160
%e A276588    261,  1370,  8742,  65304,  557400,  5343120,  56775600,  661933440
%e A276588   1631, 10112, 74046, 622704, 5900520, 62118720, 718709040, 9059339520
%t A276588 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 *)
%o A276588 (Scheme)
%o A276588 (define (A276588 n) (A276588bi (A002262 n) (A025581 n)))
%o A276588 (define (A276588bi row col) (A276075 (A066117bi (+ 1 row) (+ 1 col)))) ;; Code for A066117bi given in A066117, and for A276075 under the respective entry.
%o A276588 (PARI) T(r, c) = sum(k=0, r, binomial(r, k)*(1 + c + k)!);
%o A276588 for(r=0, 10, for(c=0, r, print1(T(c, r - c),", ");); print();) \\ _Indranil Ghosh_, Apr 11 2017
%o A276588 (Python)
%o A276588 from sympy import binomial, factorial
%o A276588 def T(r, c): return sum([binomial(r, k) * factorial(1 + c + k) for k in range(r + 1)])
%o A276588 for r in range(11): print([T(c, r - c) for c in range(r + 1)]) # _Indranil Ghosh_, Apr 11 2017
%Y A276588 Transpose: A276589.
%Y A276588 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).
%Y A276588 Leftmost column (column 0): A001339, Column 1: A001340, Columns 2-3: A001341 & A001342 (apparently).
%Y A276588 Cf. A276075.
%Y A276588 Cf. also arrays A066117, A276586, A099884, A255483.
%K A276588 nonn,tabl
%O A276588 0,2
%A A276588 _Antti Karttunen_, Sep 19 2016