A378062 Array read by ascending antidiagonals: A(n, k) = (n + 1)*binomial(2*k + n - 1, k - 1)^2 / (2*k + n - 1) for k > 0, and A(n, 0) = 0.

0, 0, 1, 0, 1, 3, 0, 1, 8, 20, 0, 1, 15, 75, 175, 0, 1, 24, 189, 784, 1764, 0, 1, 35, 392, 2352, 8820, 19404, 0, 1, 48, 720, 5760, 29700, 104544, 226512, 0, 1, 63, 1215, 12375, 81675, 382239, 1288287, 2760615, 0, 1, 80, 1925, 24200, 196625, 1145144, 5010005, 16359200, 34763300

Offset: 0

Author

Peter Luschny, Dec 07 2024

Keywords

• nonn
• tabl

Examples

			Array A(n, k) starts:
  [0] 0, 1,  3,   20,   175,    1764,    19404, ... A000891
  [1] 0, 1,  8,   75,   784,    8820,   104544, ... A145600
  [2] 0, 1, 15,  189,  2352,   29700,   382239, ... A145601
  [3] 0, 1, 24,  392,  5760,   81675,  1145144, ... A145602
  [4] 0, 1, 35,  720, 12375,  196625,  3006003, ... A145603
  [5] 0, 1, 48, 1215, 24200,  429429,  7154784, ...
  [6] 0, 1, 63, 1925, 44044,  869505, 15767024, ...
  [7] 0, 1, 80, 2904, 75712, 1656200, 32626944, ...
.
Seen as a triangle, T(n, k) = A(n-k, k). Compare the descending antidiagonals of A378061.
  [0] 0;
  [1] 0, 1;
  [2] 0, 1,  3;
  [3] 0, 1,  8,  20;
  [4] 0, 1, 15,  75,  175;
  [5] 0, 1, 24, 189,  784,  1764;
  [6] 0, 1, 35, 392, 2352,  8820,  19404;
  [7] 0, 1, 48, 720, 5760, 29700, 104544, 226512;
		

Crossrefs

Rows: A000891, A145600, A145601, A145602, A145603.

Columns: A005563, A005565, A378061.

Programs

Maple

A := (n, k) -> ifelse(k = 0, 0, (n + 1)*binomial(2*k + n - 1, k - 1)^2/(2*k + n - 1)):
for n from 0 to 7 do seq(A(n, k), k = 0..7);

Mathematica

A[n_, k_] := If[k==0, 0, (n + 1)*Binomial[2*k + n - 1, k - 1]^2 / (2*k + n - 1)]; Table[A[n-k,k],{n,0,9},{k,0,n}]//Flatten (* Stefano Spezia, Dec 08 2024 *)