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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.
1
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
0,6
EXAMPLE
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;
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 *)
CROSSREFS
KEYWORD
nonn,tabl,new
AUTHOR
Peter Luschny, Dec 07 2024
STATUS
approved