A349933 Array read by ascending antidiagonals: the s-th column gives the central s-binomial coefficients.

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

Offset

0,5

Links

Table of n, a(n) for n=0..65.

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).

Cf. A007318, A008287, A025012, A027907, A035343, A063260.

Cf. A349934, A349935.

Sequence in context: A332403 A263341 A201198 * A120258 A201922 A181644

Adjacent sequences: A349930 A349931 A349932 * A349934 A349935 A349936

Keyword

nonn,easy,tabl

Author

Stefano Spezia, Dec 06 2021

Status

approved