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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
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. A349933.
Sequence in context: A048471 A067345 A242431 * A188416 A361654 A160185
KEYWORD
nonn,easy,tabl
AUTHOR
Stefano Spezia, Dec 06 2021
STATUS
approved