A269730 Dimensions of the 2-polytridendriform operad TDendr_2.

1, 5, 31, 215, 1597, 12425, 99955, 824675, 6939769, 59334605, 513972967, 4501041935, 39784038517, 354455513105, 3179928556219, 28701561707675, 260447708523505, 2374690737067925, 21744508765633327, 199877846477679815, 1843718766426242221, 17060955558786455705, 158333204443000060291

Offset: 1

Author

N. J. A. Sloane, Mar 08 2016

Keywords

• nonn
• easy

Links

• Gheorghe Coserea, Table of n, a(n) for n = 1..512
• Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016; Adv. Appl. Math., 77, 3-85, 2016.

Crossrefs

Cf. A001047, A001263, A126216, A269731, A269732, A001003, A006442.

Programs

Mathematica

Rest[CoefficientList[Series[(1 - 5*x - Sqrt[1 - 10*x + x^2])/(12*x), {x, 0, 20}], x]] (* Vaclav Kotesovec, Apr 24 2016 *)
Table[-I*LegendreP[n, -1, 2, 5]/Sqrt[6], {n, 1, 20}] (* Vaclav Kotesovec, Apr 24 2016 *)

PARI

A001263(n,k) = binomial(n-1,k-1) * binomial(n, k-1)/k;
dimTDendr(n,q) = sum(k = 0, n-1, (q+1)^k * q^(n-k-1) * A001263(n,k+1));
my(q=2); vector(23, n, dimTDendr(n,q)) \\ Gheorghe Coserea, Apr 23 2016

PARI

my(q=2, x='x + O('x^24)); Vec(serreverse(x/((1+q*x)*(1+(q+1)*x)))) \\ Gheorghe Coserea, Sep 30 2017

Formula

a(n) = P_n(2), where P_n(x) is the polynomial associated with row n of triangle A126216 in order of decreasing powers of x.

Recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Apr 24 2016

a(n) ~ sqrt(12 + 5*sqrt(6)) * (5 + 2*sqrt(6))^n / (12*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 24 2016

A(x) = -serreverse(A001047(x))(-x). - Gheorghe Coserea, Sep 30 2017

From Peter Bala, Dec 25 2020: (Start)

a(n) = (1/(2*m*(m+1))) * Integral_{x = 1..2*m+1} Legendre_P(n,x) dx at m = 2.

a(n) = (1/(2*n+1)) * (1/(2*m*(m+1))) * ( Legendre_P(n+1,2*m+1) - Legendre_P(n-1,2*m+1) ) at m = 2. (End)

G.f. A(x) = x*exp( Sum_{n >= 1} A006442(n)*x^n/n ). - Peter Bala, Jan 09 2022

Extensions

More terms from Gheorghe Coserea, Apr 23 2016