A005721 -id:A005721 - cp's OEIS frontend

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^(2j-1) + x_j^(-(2j-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

Seiichi Manyama, Nov 02 2019

			(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, ...

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

Columns k=0-3 give A000007, A000984, A005721, A063419.
Rows n=0-2 give A000012, A005843, 2*A143166.
Main diagonal gives A329021.
Cf. A077042.

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 *)

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.

A380899 Three-Catalan Triangle read by rows, for n>=0 and k>=0.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 9, 11, 10, 6, 3, 1, 34, 90, 120, 120, 96, 64, 35, 15, 5, 1, 364, 1000, 1400, 1505, 1351, 1044, 700, 406, 202, 84, 28, 7, 1, 4269, 11925, 17225, 19425, 18657, 15753, 11845, 7965, 4785, 2553, 1197, 485, 165, 45, 9, 1

Offset: 0

Michel Marcus, Feb 07 2025

			Triangle begins:
   1
   1    1    1    1
   4    9   11   10    6    3   1
  34   90  120  120   96   64  35  15   5  1
 364 1000 1400 1505 1351 1044 700 406 202 84 28 7 1
 ...

Boualam Rezig and Moussa Ahmia, Combinatorics of three-Catalan numbers and some positivities, arXiv:2502.03615 [math.CO], 2025. See Table 2 p. 5. T(4,7)=405 is a typo.

Row sums are A005721.

Cf. A008287.

t(n, k) = polcoef((1 + x + x^2 + x^3)^n, k); \\ A008287
T(n, k) = t(2*n, 3*n+k) - t(2*n, 3*n+k+1);
row(n) = vector(3*n+1, k, T(n,k-1));

T(n, k) = A008287(2*n, 3*n+k) - A008287(2*n, 3*n+k+1).

cp's OEIS Frontend

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^(2j-1) + x_j^(-(2j-1)) )^(2*n).

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A380899 Three-Catalan Triangle read by rows, for n>=0 and k>=0.

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cp's OEIS Frontend

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).

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A380899 Three-Catalan Triangle read by rows, for n>=0 and k>=0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

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^(2j-1) + x_j^(-(2j-1)) )^(2*n).