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A127897
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Series reversion of x/(1 + 2*x + 3*x^2 + x^3).
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14
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0, 1, 2, 7, 27, 114, 507, 2342, 11125, 54002, 266684, 1335610, 6767477, 34629709, 178701317, 928903447, 4859345882, 25563551782, 135153617840, 717740916202, 3826894116962, 20478451476328, 109945087353190, 592048943478464, 3196930550222605, 17306392059508743, 93905862139673832
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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G.f.: 2*sqrt(3)*sqrt((1+x)/x)*sin(arcsin(3*sqrt(3)/(2*sqrt((1+x)/x)))/3)/3;
a(n) = Sum_{k=0..n-1} Sum_{j=0..k} (1/(2k+j-1))*C(n-1,3k-j)*C(3k-j,k)*C(k,j)*2^(n-3k+j-1)*3^j;
Recurrence: 2*n*(2*n+1)*a(n) = (3*n-1)*(5*n-2)*a(n-1) + 2*(n-2)*(21*n-20)*a(n-2) + 23*(n-3)*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
G.f.: Sum_{n>=1} binomial(3*n, n-1)/n * x^n / (1+x)^n. - Paul D. Hanna, Feb 04 2018
G.f. A(x) satisfies: A(x) = x * (1 + 2*A(x) + 3*A(x)^2 + A(x)^3). - Ilya Gutkovskiy, Jul 01 2020
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MATHEMATICA
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Flatten[{0, Rest[CoefficientList[Series[2*Sqrt[3]*Sqrt[(1+x)/x]*Sin[ArcSin[3*Sqrt[3]/(2*Sqrt[(1+x)/x])]/3]/3, {x, 0, 20}], x]]}] (* Vaclav Kotesovec, Oct 20 2012 *)
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PROG
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(PARI) {a(n) = my(A = sum(m=1, n, binomial(3*m, m-1)/m * x^m / (1+x +x*O(x^n))^m ) ); polcoeff(A, n)}
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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