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A349934 Array read by ascending antidiagonals: A(n, s) is the n-th s-Catalan number. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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 A160185 A283424

Adjacent sequences:  A349931 A349932 A349933 * A349935 A349936 A349937

KEYWORD

nonn,easy,tabl

AUTHOR

Stefano Spezia, Dec 06 2021

STATUS

approved

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Last modified August 16 20:45 EDT 2022. Contains 356169 sequences. (Running on oeis4.)