|
%I #21 Oct 11 2017 05:14:37
%S 1,2,10,62,422,2992,21736,160442,1197798,9018656,68355820,520851212,
%T 3986036204,30615867128,235879185188,1822138940482,14108173076358,
%U 109454660444336,850687921793836,6622072711690452
%N 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.
%H G. C. Greubel, <a href="/A243034/b243034.txt">Table of n, a(n) for n = 0..1000</a>
%F 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))).
%F G.f.: A(x) = x*F'(x)/(F(x)-F(x)^2), where F(x)/x is g.f. of A186997.
%F a(n) ~ (3+5*sqrt(3)) * 8^n / (33*sqrt(Pi*n)). - _Vaclav Kotesovec_, May 31 2014
%t 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_ *)
%o (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);
%o (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
%Y Cf. A186997.
%K nonn
%O 0,2
%A _Vladimir Kruchinin_, May 29 2014
|
