A050159 -id:A050159 - cp's OEIS frontend

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

Paul Barry, Dec 08 2004

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

Cf. A050159, A101601, A101602, A098658.

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

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

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

A101601 G.f.: c(3x)^3, c(x) the g.f. of A000108.

Original entry on oeis.org

1, 9, 81, 756, 7290, 72171, 729729, 7505784, 78298974, 826489170, 8811646074, 94753804536, 1026499549140, 11192793160815, 122744496427425, 1352917116177840, 14979996753469110, 166542316847391870, 1858400773709785470, 20806975169765062200, 233671377667405024620

Offset: 0

Paul Barry, Dec 08 2004

Cf. A050159, A101600, A101602.

terms = 18;
c[x_] = (1 - Sqrt[1 - 4x])/(2x) + O[x]^terms // Normal;
CoefficientList[c[3x]^3, x][[1 ;; terms]] (* Jean-François Alcover, Dec 15 2017 *)

G.f.: 8/(1+sqrt(1-12*x))^3.
a(n) = (3*n+3)/(n+3) * 3^n * C(n+1).
Conjecture: (n+3)*a(n) -3*(5*n+7)*a(n-1) +18*(2*n-1)*a(n-2)=0. - R. J. Mathar, Nov 15 2011
From Amiram Eldar, May 15 2022: (Start)
Sum_{n>=0} 1/a(n) = 51/121 + 964*arcsin(1/(2*sqrt(3)))/(121*sqrt(11)).
Sum_{n>=0} (-1)^n/a(n) = 57/169 + 1204*arcsinh(1/(2*sqrt(3)))/(169*sqrt(13)). (End)

A101602 G.f.: c(3x)^4, c(x) the g.f. of A000108.

Original entry on oeis.org

1, 12, 126, 1296, 13365, 138996, 1459458, 15466464, 165297834, 1780130520, 19301700924, 210564010080, 2309623985565, 25458117777540, 281857732537050, 3133071216411840, 34953325758094590, 391242268149428520, 4392583646950402020, 49454259823789423200

Offset: 0

Paul Barry, Dec 08 2004

Cf. A000108, A050159, A101600, A101601.

a[n_] := ((8*n + 12)/(3*n + 12)) * ((3*n + 3)/(n + 3))* 3^n* CatalanNumber[n + 1]; Array[a, 20, 0] (* Amiram Eldar, May 15 2022 *)

G.f.: 16/(1+sqrt(1-12x))^4.
a(n)=((8n+12)/(3n+12))((3n+3)/(n+3))3^n*C(n+1).
Conjecture: (n+4)*a(n) - 6*(3*n+7)*a(n-1) + 36*(2*n+1)*a(n-2) = 0. - R. J. Mathar, Nov 15 2011
From Amiram Eldar, May 15 2022: (Start)
Sum_{n>=0} 1/a(n) = 1479/484 - 2691*arcsin(1/(2*sqrt(3)))/(121*sqrt(11)).
Sum_{n>=0} (-1)^n/a(n) = 7569*arcsinh(1/(2*sqrt(3)))/(169*sqrt(13)) - 1767/676. (End)

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A101600 Expansion of g.f. c(3x)^2, where c(x) is the g.f. of A000108.

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A101601 G.f.: c(3x)^3, c(x) the g.f. of A000108.

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A101602 G.f.: c(3x)^4, c(x) the g.f. of A000108.

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