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A375177

a(n) = 1/(n + 1)^2 * Sum_{k = 1..n+1} (k^4)*binomial(n+1, k)^2.

1, 5, 26, 134, 670, 3262, 15540, 72732, 335478, 1528670, 6894316, 30820660, 136736236, 602610764, 2640266600, 11508115320, 49928451750, 215717144670, 928515985980, 3983029119300, 17032882625220, 72631992447300, 308911087394520, 1310670689270280, 5548646191175100

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OFFSET

1,2

COMMENTS

Compare with the identity 1/(n+1)^2 * Sum_{k = 1..n+1} (k^2)*binomial(n+1, k)^2 = binomial(2*n, n) = A000984(n).

The central binomial coefficients satisfy the supercongruence binomial(2*p, p) == 2 (mod p^3) for all primes p >= 5 (Wolstenholme's theorem).

LINKS

Table of n, a(n) for n=1..25.

Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2012), arXiv:1111.3057 [math.NT], (2011).

FORMULA

a(n) = hypergeom([2, 2, -n, -n], [1, 1, 1], 1).

a(n) = (1/2) * P(n)/(2*n - 1) * binomial(2*n, n), where P(n) = n^3 + 4*n^2 + 2*n - 2.