A379082 Expansion of (1/x) * Series_Reversion( x * (1/(1 + x) - x^3)^2 ).
1, 2, 5, 16, 64, 288, 1354, 6496, 31728, 157818, 798098, 4091712, 21211165, 110969430, 585116287, 3106334810, 16590881379, 89085610328, 480627775528, 2604103448334, 14163573236255, 77302955664902, 423245859576867, 2324046398587426, 12795255089638583, 70617777139027756
Offset: 0
Keywords
Programs
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PARI
a(n) = 2*sum(k=0, n\3, binomial(2*n+k+2, k)*binomial(2*n+k+2, n-3*k)/(2*n+k+2));
Formula
G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{k>=1} A379085(k) * x^k/k ).
(2) A(x) = ( (1 + x*A(x)) * (1 + x^3*A(x)^(7/2)) )^2.
(3) A(x) = B(x)^2 where B(x) is the g.f. of A379089.
a(n) = (1/(n+1)) * [x^n] 1/( 1/(1 + x) - x^3 )^(2*(n+1)).
a(n) = 2 * Sum_{k=0..floor(n/3)} binomial(2*n+k+2,k) * binomial(2*n+k+2,n-3*k)/(2*n+k+2) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(2*n+k+1,k) * binomial(2*n+k+2,n-3*k).