A329020 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of ( Sum_{j=1..k} x_j^(2*j-1) + x_j^(-(2*j-1)) )^(2*n).

1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 44, 20, 0, 1, 8, 146, 580, 70, 0, 1, 10, 344, 4332, 8092, 252, 0, 1, 12, 670, 18152, 135954, 116304, 924, 0, 1, 14, 1156, 55252, 1012664, 4395456, 1703636, 3432, 0, 1, 16, 1834, 137292, 4816030, 58199208, 144840476, 25288120, 12870, 0

Offset

0,5

Links

Seiichi Manyama, Antidiagonals n = 0..50, flattened

Formula

T(n,k) = Sum_{j=0..floor((2*k-1)*n/(2*k))} (-1)^j * binomial(2*n,j) * binomial((2*k+1)*n-2*k*j-1,(2*k-1)*n-2*k*j) for k > 0.

Example

(x^3 + x + 1/x + 1/x^3)^2 = x^6 + 2*x^4 + 3*x^2 + 4 + 3/x^2 + 2/x^4 + 1/x^6. So T(1,2) = 4.
Square array begins:
   1,   1,      1,       1,        1,         1, ...
   0,   2,      4,       6,        8,        10, ...
   0,   6,     44,     146,      344,       670, ...
   0,  20,    580,    4332,    18152,     55252, ...
   0,  70,   8092,  135954,  1012664,   4816030, ...
   0, 252, 116304, 4395456, 58199208, 432457640, ...

Mathematica

T[n_, 0] = Boole[n == 0]; T[n_, k_] := Sum[(-1)^j * Binomial[2*n, j] * Binomial[(2*k + 1)*n - 2*k*j - 1, (2*k - 1)*n - 2*k*j], {j, 0, Floor[(2*k - 1)*n/(2*k)]}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 06 2021 *)

Crossrefs

Columns k=0-3 give A000007, A000984, A005721, A063419.

Rows n=0-2 give A000012, A005843, 2*A143166.

Main diagonal gives A329021.

Cf. A077042.

Sequence in context: A106579 A378318 A287318 * A351640 A173003 A378236

Adjacent sequences: A329017 A329018 A329019 * A329021 A329022 A329023

Keyword

nonn,tabl

Author

Seiichi Manyama, Nov 02 2019

Status

approved