A381775 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^6 ) )^(1/6), where C(x) is the g.f. of A000108.
1, 2, 27, 523, 11871, 294668, 7747698, 212054604, 5978347887, 172421233231, 5063192676597, 150872475295522, 4550458484780442, 138652322209300991, 4261638256558924407, 131973650298641750844, 4113788296015093994719, 128973000885015536107140
Offset: 0
Keywords
Programs
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PARI
my(N=20, x='x+O('x^N)); Vec((serreverse(x/((1+x)*(1-sqrt(1-4*x))/(2*x))^6)/x)^(1/6))
Formula
G.f. A(x) satisfies A(x) = (1 + x*A(x)^6) * C(x*A(x)^6).
a(n) = Sum_{k=0..n} binomial(6*n+2*k+1,k) * binomial(6*n+1,n-k)/(6*n+2*k+1).
a(n) = binomial(1 + 6*n, n)*hypergeom([-n, 1/2+3*n, 1+3*n], [2+5*n, 2+6*n], -4)/(1 + 6*n). - Stefano Spezia, Mar 07 2025