A238696 a(n) = Sum_{k=0..floor(n/2)} binomial(n*(n-k), n*k).

1, 1, 2, 21, 497, 18508, 3297933, 2348121769, 2319121509374, 4535739243360613, 58887253765506968848, 1694438232474931034462251, 64598311562133275526222276162, 8312693334404799592869803398802772, 5827069387752679429926992257426553147833

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..69

Vaclav Kotesovec, Limits, graph for 500 terms

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

Cf. A206849, A207136, A209331.

Sequence in context: A090451 A199747 A303867 * A226057 A158886 A092957

Adjacent sequences: A238693 A238694 A238695 * A238697 A238698 A238699

Keyword

nonn,nice

Author

Vaclav Kotesovec, Mar 03 2014

Status

approved