A101600 Expansion of g.f. c(3x)^2, where c(x) is the g.f. of A000108.
1, 6, 45, 378, 3402, 32076, 312741, 3127410, 31899582, 330595668, 3471254514, 36848701764, 394807518900, 4263921204120, 46370143094805, 507343918566690, 5580783104233590, 61682339573108100, 684673969261499910, 7629224228913856140, 85308598196036755020
Offset: 0
Links
- Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016.
Programs
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Maple
Z[0]:=1: for k to 30 do Z[k]:=simplify(1/(1-3*z*Z[k-1])) od: g:=sum((Z[j]-Z[j-1]), j=1..30): gser:=series(g, z=0, 27): seq(coeff(gser, z, n)/3, n=1..19); # Zerinvary Lajos, May 21 2008
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Mathematica
a[n_] := 3^n * CatalanNumber[n + 1]; Array[a, 20, 0] (* Amiram Eldar, May 15 2022 *)
Formula
G.f.: 4/(1+sqrt(1-12*x))^2.
a(n) = 3^n * A000108(n+1).
(n+2)*a(n) -6*(2*n+1)*a(n-1)=0. - R. J. Mathar, Nov 15 2011
O.g.f. A(x) = 1/x*series reversion( x/(1 + 3*x)^2 ). 1 + x*A'(x)/A(x) = 1/sqrt(1 - 12*x) is the o.g.f. for A098658. - Peter Bala, Jul 17 2015
From Amiram Eldar, May 15 2022: (Start)
Sum_{n>=0} 1/a(n) = 87/121 + 648*arcsin(1/(2*sqrt(3)))/(121*sqrt(11)).
Sum_{n>=0} (-1)^n/a(n) = 93/169 + 648*arcsinh(1/(2*sqrt(3)))/(169*sqrt(13)). (End)
E.g.f.: BesselI(1,6*z)*exp(6*z)/(3*z) where BesselI is the modified Bessel function of type I. - Karol A. Penson, Feb 17 2025