A101600 G.f.: c(3x)^2, c(x) 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

0,2

Links

Table of n, a(n) for n=0..20.

Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016.

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)

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

Mathematica

a[n_] := 3^n * CatalanNumber[n + 1]; Array[a, 20, 0] (* Amiram Eldar, May 15 2022 *)

Crossrefs

Cf. A050159, A101601, A101602, A098658.

Sequence in context: A153399 A007194 A025551 * A233668 A243694 A227169

Adjacent sequences: A101597 A101598 A101599 * A101601 A101602 A101603

Keyword

easy,nonn

Author

Paul Barry, Dec 08 2004

Status

approved