A349935 Array read by ascending antidiagonals: A(n, k) is the n-th spin s-Catalan number, with s = k/2.

1, 0, 1, 2, 1, 1, 0, 3, 0, 1, 5, 6, 4, 1, 1, 0, 15, 0, 5, 0, 1, 14, 36, 34, 16, 6, 1, 1, 0, 91, 0, 65, 0, 7, 0, 1, 42, 232, 364, 260, 111, 31, 8, 1, 1, 0, 603, 0, 1085, 0, 175, 0, 9, 0, 1, 132, 1585, 4269, 4600, 2666, 981, 260, 51, 10, 1, 1, 0, 4213, 0, 19845, 0, 5719, 0, 369, 0, 11, 0, 1

Offset

1,4

Links

Table of n, a(n) for n=1..78.

William Linz, s-Catalan numbers and Littlewood-Richardson polynomials, arXiv:2110.12095 [math.CO], 2021. See p. 3.

Formula

A(n, k) = T(n, k*(n+1)/2, k) - T(n, k*(n+1)/2+1, k), 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(n, 1) = A126120(n+1).

A(n, 2) = A005043(n+1).

A(3, n) = A000027(n+1).

A(4, 2*n) = A005891(n).

A(5, n) = A006003(n+1).

Example

The array begins:
n\k | 1    2    3    4    5    6
----+---------------------------
  1 | 1    1    1    1    1    1 ...
  2 | 0    1    0    1    0    1 ...
  3 | 2    3    4    5    6    7 ...
  4 | 0    6    0   16    0   31 ...
  5 | 5   15   34   65  111  175 ...
  6 | 0   36    0  260    0  981 ...
  ...

Mathematica

T[n_, k_, s_]:=If[k==0, 1, Coefficient[(Sum[x^i, {i, 0, s}])^n, x^k]]; A[n_, k_]:=T[n, k(n+1)/2, k]-T[n, k(n+1)/2+1, k]; Flatten[Table[A[n-k+1, k], {n, 12}, {k, n}]]

Crossrefs

Cf. A000012 (1st row), A059841 (2nd row).

Cf. A000027, A005043, A005891, A006003, A126120.

Cf. A349934.

Sequence in context: A322435 A113949 A318808 * A257991 A343029 A343037

Adjacent sequences: A349932 A349933 A349934 * A349936 A349937 A349938

Keyword

nonn,easy,tabl

Author

Stefano Spezia, Dec 06 2021

Status

approved