A127897 Series reversion of x/(1 + 2*x + 3*x^2 + x^3).

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

Offset: 0

Author

Paul Barry, Feb 04 2007

Keywords

• easy
• nonn

Comments

Series reversion of A127896.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100
• Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.

Programs

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 *)

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

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