OFFSET
1,1
COMMENTS
The second program in Mathematica is not suitable for calculating more than 99 terms, as there are unevaluated terms with HypergeometricU. - Vaclav Kotesovec, Feb 05 2026
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..280 (terms 1..100 from Andrew Howroyd)
Eric Weisstein's World of Mathematics, Complete Tripartite Graph
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Matching
FORMULA
a(n) = Sum_{i,j,k} binomial(n, i)^2 * binomial(n, j) * binomial(n-i, j) * binomial(n-i, k) * binomial(n-j, k) * i!*j!*k!. - Andrew Howroyd, Oct 02 2017
From Vaclav Kotesovec, Feb 05 2026: (Start)
Recurrence: (2*n - 1)*a(n) = -(2*n^3 - 15*n^2 + 7*n + 2)*a(n-1) + (n-1)^2*(28*n^2 - 28*n - 15)*a(n-2) + 8*(n-2)^3*(n-1)^2*(2*n + 1)*a(n-3).
a(n) ~ 2^((3*n+1)/2) * exp(3*(sqrt(n/2) - n/2 - 1/8)) * n^(3*n/2) * (1 - 11/(2^(9/2)*sqrt(n))). (End)
MATHEMATICA
Table[Sum[Binomial[n, i]^2 Binomial[n, j] Binomial[n - i, j] Binomial[n - i, k] Binomial[n - j, k] i! j! k!, {i, 0, n}, {j, 0, n - i}, {k, 0, Min[n - i, n - j]}], {n, 20}]
Table[Sum[(-1)^(i - n) Binomial[n, i]^2 Binomial[n, j] Binomial[-i + n, j] i! j! HypergeometricU[i - n, 1 + i - j, -1], {i, 0, n}, {j, 0, n - i}], {n, 20}]
PROG
(PARI) a(n) = sum(i=0, n, sum(j=0, n-i, sum(k=0, min(n-i, n-j), binomial(n, i)^2 * binomial(n, j) * binomial(n-i, j) * binomial(n-i, k) * binomial(n-j, k) * i!*j!*k!))); \\ Andrew Howroyd, Oct 02 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Sep 30 2017
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Oct 02 2017
STATUS
approved