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A349933 Array read by ascending antidiagonals: the s-th column gives the central s-binomial coefficients. 3
1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 20, 19, 4, 1, 1, 70, 141, 44, 5, 1, 1, 252, 1107, 580, 85, 6, 1, 1, 924, 8953, 8092, 1751, 146, 7, 1, 1, 3432, 73789, 116304, 38165, 4332, 231, 8, 1, 1, 12870, 616227, 1703636, 856945, 135954, 9331, 344, 9, 1, 1, 48620, 5196627, 25288120, 19611175, 4395456, 398567, 18152, 489, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
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), 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.
EXAMPLE
The array begins:
n\s | 0 1 2 3 4
----+----------------------------
0 | 1 1 1 1 1 ...
1 | 1 2 3 4 5 ...
2 | 1 6 19 44 85 ...
3 | 1 20 141 580 1751 ...
4 | 1 70 1107 8092 38165 ...
...
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]; Flatten[Table[A[n-s, s], {n, 0, 9}, {s, 0, n}]]
CROSSREFS
Cf. A000984 (s=1), A082758 (s=2), A005721 (s=3), A349936 (s=4), A063419 (s=5), A270918 (n=s), A163269 (s>0).
Sequence in context: A332403 A263341 A201198 * A120258 A201922 A181644
KEYWORD
nonn,easy,tabl
AUTHOR
Stefano Spezia, Dec 06 2021
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)