A371404 Expansion of (1/x) * Series_Reversion( x / ( (1+x) * (1+3*x)^2 ) ).
1, 7, 64, 667, 7513, 89092, 1095832, 13852195, 178855075, 2348744095, 31273438804, 421224534100, 5728966150924, 78569975545432, 1085350298162608, 15087689038165555, 210907141968410071, 2962825568825439349, 41806163408065511032, 592244891188614804643
Offset: 0
Programs
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Maple
seq(simplify(hypergeom([-n, -2*(n+1)], [2], 3)), n = 0..20); # Peter Bala, Sep 08 2024
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PARI
my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)*(1+3*x)^2))/x) -
PARI
a(n) = sum(k=0, n, 3^k*binomial(2*(n+1), k)*binomial(n+1, n-k))/(n+1);
Formula
a(n) = (1/(n+1)) * Sum_{k=0..n} 3^k * binomial(2*(n+1),k) * binomial(n+1,n-k).
From Peter Bala, Sep 08 2024: (Start)
a(n) = hypergeom([-n, -2*(n+1)], [2], 3).
a(n) = (-2)^n * Jacobi_P(n, 1, n+2, -2)/(n+1).
P-recursive: 2*(11*n-5)*(2*n+3)*(n+1)*a(n) = (649*n^3+354*n^2-109*n-54)*a(n-1) + 16*(n-1)*(2*n-1)*(11*n+6)*a(n-2) with a(0) = 1 and a(1) = 7. (End)