A372109 G.f. A(x) satisfies A(x) = ( (1 - x*A(x))/(1 - 5*x*A(x)) )^(1/2).
1, 2, 12, 90, 758, 6850, 64904, 636250, 6399120, 65661250, 684665828, 7233956250, 77278356246, 833291781250, 9057750917944, 99144375156250, 1091857567068742, 12089416175781250, 134501879883249300, 1502857085910156250, 16857310306553767026
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, 4^k*binomial(n/2+k-1/2, k)*binomial(n-1, n-k))/(n+1);
Formula
a(n) = (1/(n+1)) * Sum_{k=0..n} 4^k * binomial(n/2+k-1/2,k) * binomial(n-1,n-k).
From Seiichi Manyama, Nov 30 2024: (Start)
G.f.: exp( Sum_{k>=1} A378551(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] 1/(1 - 4*x/(1-x))^((n+1)/2).
G.f.: (1/x) * Series_Reversion( x*(1 - 4*x/(1-x))^(1/2) ). (End)