A378427 Expansion of (1/x) * Series_Reversion( x / (1 + x + x^3 * (1 + x)^3) ).
1, 1, 1, 2, 8, 29, 88, 253, 775, 2575, 8797, 29833, 100635, 342408, 1181727, 4120223, 14435969, 50738813, 179038408, 634696939, 2259677734, 8072923814, 28924907573, 103915759961, 374302237154, 1351541722226, 4891132336481, 17736792240766, 64440831300682
Offset: 0
Keywords
Programs
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PARI
my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x+x^3*(1+x)^3))/x) -
PARI
a(n) = sum(k=0, n\3, binomial(n+1, k)*binomial(n+2*k+1, n-3*k))/(n+1);
Formula
G.f.: exp( Sum_{k>=1} A378407(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] (1 + x + x^3 * (1 + x)^3)^(n+1).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(n+1,k) * binomial(n+2*k+1,n-3*k).