A256462 Double sum of the product of two binomials with even arguments.

1, 14, 238, 4092, 70070, 1192516, 20177164, 339653880, 5692584870, 95050101300, 1581953021220, 26255495764680, 434697697648188, 7181635051211432, 118422830335911640, 1949458102569167344, 32043155434056810246, 525974795270875804308

Offset

0,2

Comments

Summing the product binomial(k + j, j) * binomial(2n - k - j, n - j), without even arguments restriction, one gets the sequence A002457.

Links

Table of n, a(n) for n=0..17.

MathOverflow, Sums of binomials with even coefficients, 2011.

Formula

a(n) = Sum_{k = 0..n} (Sum_{j = 0..n} binomial(2*k + 2*j, 2*j) * binomial(4*n - 2*k - 2*j, 2*n - 2*j)).

a(n) = (2*n^2 + 4*n + 1)*C(2*n), where C(n) is the n-th Catalan number. In other words, a(n) = A056220(n+1)*A000108(2*n).

a(n) ~ 2^(4*n-1/2) * sqrt(n/Pi). - Amiram Eldar, Oct 10 2025

Example

For n = 2, the sum ((70+15+1) + (15+36+15) + (1+15+70)) evaluates to a(2) = 238.

Mathematica

Table[(2*n^2 + 4*n + 1)*CatalanNumber[2*n], {n, 0, 25}]

Prog

(Magma) [(2*n^2 + 4*n + 1)*Catalan(2*n): n in [0..20]]; // Vincenzo Librandi, Apr 01 2015

Crossrefs

Cf. A000108, A002457, A056220.

Sequence in context: A222377 A386900 A220502 * A166774 A286882 A326651

Adjacent sequences: A256459 A256460 A256461 * A256463 A256464 A256465

Keyword

easy,nonn

Author

Jean-François Alcover, Mar 30 2015

Status

approved