A372125 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)*(1 + 4*x*A(x))^(1/2) ).
1, 1, 4, 13, 60, 256, 1252, 5979, 30360, 153626, 801632, 4197284, 22355788, 119695396, 647666544, 3522773337, 19298660772, 106213538104, 587632185580, 3264011196578, 18203515158400, 101862717712340, 571859834176800, 3219573318768300, 18175140989890716
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, 4^(n-k)*binomial(n+k, k)*binomial(k/2, n-k))/(n+1);
Formula
a(n) = (1/(n+1)) * Sum_{k=0..n} 4^(n-k) * binomial(n+k,k) * binomial(k/2,n-k).
From Seiichi Manyama, Nov 30 2024: (Start)
G.f.: exp( Sum_{k>=1} A378554(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] 1/(1 - x*(1 + 4*x)^(1/2))^(n+1).
G.f.: (1/x) * Series_Reversion( x*(1 - x*(1 + 4*x)^(1/2)) ). (End)