A379080 Expansion of (1/x) * Series_Reversion( x * (1/(1 + x) - x^2)^2 ).
1, 2, 7, 32, 163, 886, 5039, 29616, 178446, 1096356, 6842452, 43259122, 276462247, 1783114592, 11591769207, 75874998822, 499643588823, 3307746965238, 22001986381873, 146972401234478, 985535271867577, 6631547191254298, 44763982636889092, 303037237861086682
Offset: 0
Keywords
Programs
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PARI
a(n) = 2*sum(k=0, n\2, binomial(2*n+k+2, k)*binomial(2*n+k+2, n-2*k)/(2*n+k+2));
Formula
G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{k>=1} A379084(k) * x^k/k ).
(2) A(x) = ( (1 + x*A(x)) * (1 + x^2*A(x)^(5/2)) )^2.
(3) A(x) = B(x)^2 where B(x) is the g.f. of A200719.
a(n) = (1/(n+1)) * [x^n] 1/( 1/(1 + x) - x^2 )^(2*(n+1)).
a(n) = 2 * Sum_{k=0..floor(n/2)} binomial(2*n+k+2,k) * binomial(2*n+k+2,n-2*k)/(2*n+k+2) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(2*n+k+2,n-2*k).