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A349935
Array read by ascending antidiagonals: A(n, k) is the n-th spin s-Catalan number, with s = k/2.
1
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
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. A349934.
Sequence in context: A322435 A113949 A318808 * A257991 A373592 A343029
KEYWORD
nonn,easy,tabl
AUTHOR
Stefano Spezia, Dec 06 2021
STATUS
approved