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A127897 Series reversion of x/(1 + 2*x + 3*x^2 + x^3). 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Series reversion of A127896.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..100

FORMULA

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

a(n) ~ 23^(n+1/2)/(12*4^n*sqrt(Pi)*n^(3/2)). - 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

MATHEMATICA

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 *)

PROG

(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)}

for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Feb 04 2018

CROSSREFS

Sequence in context: A150627 A150628 A106225 * A180473 A154108 A011965

Adjacent sequences:  A127894 A127895 A127896 * A127898 A127899 A127900

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Feb 04 2007

STATUS

approved

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Last modified June 15 04:47 EDT 2021. Contains 345043 sequences. (Running on oeis4.)