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

Author

Seiichi Manyama, Dec 15 2018

Keywords

• nonn
• tabl

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

Columns 0-5: A000012, A000004, A006331, A303916, A305167, A318119.

Main diagonal gives A318793.

Cf. A201552.

Programs

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