A322549 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(n,k) is the constant term in the expansion of (Sum_{j=0..n} j*(x^j + x^(-j)))^k.
1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 10, 0, 1, 0, 6, 12, 28, 0, 1, 0, 0, 198, 84, 60, 0, 1, 0, 20, 560, 2076, 324, 110, 0, 1, 0, 0, 5020, 14240, 12060, 924, 182, 0, 1, 0, 70, 20580, 213460, 146680, 49170, 2184, 280, 0, 1, 0, 0, 144774, 1984584, 3479700, 922680, 158418, 4536, 408, 0, 1
Offset: 0
Examples
Square array begins: 1, 0, 0, 0, 0, 0, 0, ... 1, 0, 2, 0, 6, 0, 20, ... 1, 0, 10, 12, 198, 560, 5020, ... 1, 0, 28, 84, 2076, 14240, 213460, ... 1, 0, 60, 324, 12060, 146680, 3479700, ... 1, 0, 110, 924, 49170, 922680, 32108060, ... 1, 0, 182, 2184, 158418, 4226040, 203474180, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..50, flattened
Crossrefs
Programs
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Mathematica
A[0, 0] = 1; A[n_, k_] := Coefficient[Expand[Sum[j * (x^j + x^(-j)), {j, 0, n}]^k], x, 0]; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Dec 16 2018 *) -
PARI
T(n,k) = my(t=sum(j=0, n, j*(x^j + x^(-j)))^k); polcoef(numerator(t), poldegree(denominator(t))); \\ Michel Marcus, Dec 17 2018