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A238696
a(n) = Sum_{k=0..floor(n/2)} binomial(n*(n-k), n*k).
4
1, 1, 2, 21, 497, 18508, 3297933, 2348121769, 2319121509374, 4535739243360613, 58887253765506968848, 1694438232474931034462251, 64598311562133275526222276162, 8312693334404799592869803398802772, 5827069387752679429926992257426553147833
OFFSET
0,3
LINKS
FORMULA
Maximum is at k = n*(1-1/sqrt(5))/2 = 0.2763932... * n.
Limit n->infinity a(n)^(1/n^2) = (1+sqrt(5))/2.
Lim sup n->infinity a(n) / (5^(1/4)/(n*sqrt(2*Pi))*((1+sqrt(5))/2)^(n^2+1)) = JacobiTheta3(0,exp(-5*sqrt(5)/2)) = EllipticTheta[3,0,Exp[-5*Sqrt[5]/2]] = 1.007468786736926147579...
Lim inf n->infinity a(n) / (5^(1/4)/(n*sqrt(2*Pi))*((1+sqrt(5))/2)^(n^2+1)) = JacobiTheta2(0,exp(-5*sqrt(5)/2)) = EllipticTheta[2,0,Exp[-5*Sqrt[5]/2]] = 0.494414344263155315970...
a(n) = [x^(n^2)] (1-x)^(n-1)/((1-x)^n - x^(2*n)) for n > 0. - Seiichi Manyama, Oct 11 2021
MATHEMATICA
Table[Sum[Binomial[n*(n-k), n*k], {k, 0, Floor[n/2]}], {n, 0, 20}]
PROG
(PARI) a(n)=sum(k=0, n\2, binomial(n*(n-k), n*k)) \\ Charles R Greathouse IV, Jul 29 2016
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Vaclav Kotesovec, Mar 03 2014
STATUS
approved