A349934 Array read by ascending antidiagonals: A(n, s) is the n-th s-Catalan number.

1, 2, 1, 5, 3, 1, 14, 15, 4, 1, 42, 91, 34, 5, 1, 132, 603, 364, 65, 6, 1, 429, 4213, 4269, 1085, 111, 7, 1, 1430, 30537, 52844, 19845, 2666, 175, 8, 1, 4862, 227475, 679172, 383251, 70146, 5719, 260, 9, 1, 16796, 1730787, 8976188, 7687615, 1949156, 204687, 11096, 369, 10, 1

Offset

1,2

Links

Table of n, a(n) for n=1..55.

William Linz, s-Catalan numbers and Littlewood-Richardson polynomials, arXiv:2110.12095 [math.CO], 2021. See p. 2.

Formula

A(n, s) = T(2*n, s*n, s) - T(2*n, s*n+1, s), where T(n, k, s) is the s-binomial coefficient defined as the coefficient of x^k in (Sum_{i=0..s} x^i)^n.

A(2, n) = A000027(n+1).

A(3, n) = A006003(n+1).

Example

The array begins:
n\s |  1    2     3      4      5
----+----------------------------
  1 |  1    1     1      1      1 ...
  2 |  2    3     4      5      6 ...
  3 |  5   15    34     65    111 ...
  4 | 14   91   364   1085   2666 ...
  5 | 42  603  4269  19845  70146 ...
  ...

Mathematica

T[n_, k_, s_]:=If[k==0, 1, Coefficient[(Sum[x^i, {i, 0, s}])^n, x^k]]; A[n_, s_]:=T[2n, s n, s]-T[2n, s n+1, s]; Flatten[Table[A[n-s+1, s], {n, 10}, {s, n}]]

Prog

(PARI) T(n, k, s) = polcoef((sum(i=0, s, x^i))^n, k);
A(n, s) = T(2*n, s*n, s) - T(2*n, s*n+1, s); \\ Michel Marcus, Dec 10 2021

Crossrefs

Cf. A000012 (n=1), A220892 (n=4).

Cf. A000108 (s=1), A099251 (s=2), A264607 (s=3).

Cf. A000027, A007318, A008287, A027907, A035343, A063260.

Cf. A349933.

Sequence in context: A048471 A067345 A242431 * A188416 A361654 A160185

Adjacent sequences: A349931 A349932 A349933 * A349935 A349936 A349937

Keyword

nonn,easy,tabl

Author

Stefano Spezia, Dec 06 2021

Status

approved