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A127897 Series reversion of x/(1 + 2*x + 3*x^2 + x^3). 12

%I #29 Jul 07 2021 02:00:33

%S 0,1,2,7,27,114,507,2342,11125,54002,266684,1335610,6767477,34629709,

%T 178701317,928903447,4859345882,25563551782,135153617840,717740916202,

%U 3826894116962,20478451476328,109945087353190,592048943478464,3196930550222605,17306392059508743,93905862139673832

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

%C Series reversion of A127896.

%H Vincenzo Librandi, <a href="/A127897/b127897.txt">Table of n, a(n) for n = 0..100</a>

%H Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.

%F G.f.: 2*sqrt(3)*sqrt((1+x)/x)*sin(arcsin(3*sqrt(3)/(2*sqrt((1+x)/x)))/3)/3;

%F 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;

%F 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

%F a(n) ~ 23^(n+1/2)/(12*4^n*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 20 2012

%F G.f.: Sum_{n>=1} binomial(3*n, n-1)/n * x^n / (1+x)^n. - _Paul D. Hanna_, Feb 04 2018

%F G.f. A(x) satisfies: A(x) = x * (1 + 2*A(x) + 3*A(x)^2 + A(x)^3). - _Ilya Gutkovskiy_, Jul 01 2020

%t 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 *)

%o (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)}

%o for(n=0,30,print1(a(n),", ")) \\ _Paul D. Hanna_, Feb 04 2018

%K easy,nonn

%O 0,3

%A _Paul Barry_, Feb 04 2007

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)