A243034 Expansion of A(x) = x*F'(x)/(F(x) - F(x)^2), where F(x) = (-1 - sqrt(1-8*x) + sqrt(2 + 2*sqrt(1-8*x) + 8*x))/4.

1, 2, 10, 62, 422, 2992, 21736, 160442, 1197798, 9018656, 68355820, 520851212, 3986036204, 30615867128, 235879185188, 1822138940482, 14108173076358, 109454660444336, 850687921793836, 6622072711690452, 51621868156476212, 402929115540626240, 3148664886787313728

Offset: 0

Author

Vladimir Kruchinin, May 29 2014

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A186997.

Programs

Mathematica

Table[1+n*Sum[Sum[Binomial[k, n-m-k]*Binomial[n+2*k-1, n+k-1]/(n+k), {k, 1, n-m}], {m, 0, n}],{n,0,20}] (* Vaclav Kotesovec, May 31 2014 after Vladimir Kruchinin *)

Maxima

a(n):=1+n*sum(sum((binomial(k,n-m-k)*binomial(n+2*k-1,n+k-1))/(n+k),k,1,n-m),m,0,n);

PARI

for(n=0,25, print1(1 + n*sum(m=0,n, sum(k=1,n-m, (binomial(k,n-m-k)*binomial(n+2*k-1,n+k-1))/(n+k))), ", ")) \\ G. C. Greubel, Jun 01 2017

Formula

a(n) = 1 + n*Sum_{m=0..n} Sum_{k=1..(n-m)} binomial(k, n-m-k) * binomial(n+2*k-1, n+k-1) / (n+k).

G.f.: A(x) = x*F'(x)/(F(x)-F(x)^2), where F(x)/x is g.f. of A186997.

a(n) ~ (3+5*sqrt(3)) * 8^n / (33*sqrt(Pi*n)). - Vaclav Kotesovec, May 31 2014