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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 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.)