|
%I #41 Jan 10 2022 11:43:45
%S 1,5,31,215,1597,12425,99955,824675,6939769,59334605,513972967,
%T 4501041935,39784038517,354455513105,3179928556219,28701561707675,
%U 260447708523505,2374690737067925,21744508765633327,199877846477679815,1843718766426242221,17060955558786455705,158333204443000060291
%N Dimensions of the 2-polytridendriform operad TDendr_2.
%H Gheorghe Coserea, <a href="/A269730/b269730.txt">Table of n, a(n) for n = 1..512</a>
%H Samuele Giraudo, <a href="http://arxiv.org/abs/1603.01394">Pluriassociative algebras II: The polydendriform operad and related operads</a>, arXiv:1603.01394 [math.CO], 2016; Adv. Appl. Math., 77, 3-85, 2016.
%F 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.
%F Recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - (n-2)*a(n-2). - _Vaclav Kotesovec_, Apr 24 2016
%F a(n) ~ sqrt(12 + 5*sqrt(6)) * (5 + 2*sqrt(6))^n / (12*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Apr 24 2016
%F A(x) = -serreverse(A001047(x))(-x). - _Gheorghe Coserea_, Sep 30 2017
%F From _Peter Bala_, Dec 25 2020: (Start)
%F a(n) = (1/(2*m*(m+1))) * Integral_{x = 1..2*m+1} Legendre_P(n,x) dx at m = 2.
%F 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)
%F G.f. A(x) = x*exp( Sum_{n >= 1} A006442(n)*x^n/n ). - _Peter Bala_, Jan 09 2022
%t Rest[CoefficientList[Series[(1 - 5*x - Sqrt[1 - 10*x + x^2])/(12*x), {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Apr 24 2016 *)
%t Table[-I*LegendreP[n, -1, 2, 5]/Sqrt[6], {n, 1, 20}] (* _Vaclav Kotesovec_, Apr 24 2016 *)
%o (PARI)
%o A001263(n,k) = binomial(n-1,k-1) * binomial(n, k-1)/k;
%o dimTDendr(n,q) = sum(k = 0, n-1, (q+1)^k * q^(n-k-1) * A001263(n,k+1));
%o my(q=2); vector(23, n, dimTDendr(n,q)) \\ _Gheorghe Coserea_, Apr 23 2016
%o (PARI) my(q=2, x='x + O('x^24)); Vec(serreverse(x/((1+q*x)*(1+(q+1)*x)))) \\ _Gheorghe Coserea_, Sep 30 2017
%Y Cf. A001047, A001263, A126216, A269731, A269732, A001003, A006442.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_, Mar 08 2016
%E More terms from _Gheorghe Coserea_, Apr 23 2016
|
