A378889 G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(4/3)/(1 + x*A(x)^(1/3)) )^3.
1, 3, 12, 61, 348, 2127, 13617, 90132, 611802, 4235405, 29788821, 212255520, 1528928674, 11115361491, 81452537253, 601004875689, 4461440570523, 33295962947925, 249673885001674, 1880204670772221, 14213624028779964, 107823953314047139, 820541644515512502
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Keywords
Programs
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PARI
a(n, r=3, s=-1, t=4, u=1) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));
Formula
G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)/(1 + x*A(x)^(1/3)) )^3.
G.f. A(x) satisfies A(x) = 1 + x * A(x)^(1/3) * (1 + A(x)^(4/3) + A(x)^(5/3)).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A364758.
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r).