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%I #43 Sep 08 2022 08:46:16
%S 1,7,61,595,6217,68047,770149,8939707,105843409,1273241431,
%T 15517824973,191202877411,2377843390873,29807864423071,
%U 376255282112629,4778240359795147,61007205215610529,782648075371992487,10083436451634033757,130413832663780730995,1692599303723819234281,22037570163808433691247,287762084009227350367621
%N Dimensions of the 3-polytridendriform operad TDendr_3.
%H Gheorghe Coserea, <a href="/A269731/b269731.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(3), 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) = 7*(2*n-1)*a(n-1) - (n-2)*a(n-2). - _Vaclav Kotesovec_, Apr 24 2016
%F a(n) ~ sqrt(24 + 14*sqrt(3)) * (7 + 4*sqrt(3))^n / (24*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Apr 24 2016
%F A(x) = -serreverse(A005061(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 = 3.
%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 = 3. (End)
%t Rest[CoefficientList[Series[(1 - 7*x - Sqrt[1 - 14*x + x^2])/(24*x), {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Apr 24 2016 *)
%t Table[-I*LegendreP[n, -1, 2, 7]/(2*Sqrt[3]), {n, 1, 20}] (* _Vaclav Kotesovec_, Apr 24 2016 *)
%t RecurrenceTable[{a[1] == 1, a[2] == 7, (n + 1) a[n] == 7 (2 n - 1) a[n-1] - (n - 2) a[n-2]}, a, {n, 25}] (* _Vincenzo Librandi_, Nov 29 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=3); vector(23, n, dimTDendr(n,q)) \\ _Gheorghe Coserea_, Apr 23 2016
%o (PARI) my(q=3, x='x + O('x^24)); Vec(serreverse(x/((1+q*x)*(1+(q+1)*x)))) \\ _Gheorghe Coserea_, Sep 30 2017
%o (Magma) I:=[1,7]; [n le 2 select I[n] else (7*(2*n-1)*Self(n-1)-(n-2)*Self(n-2))/(n+1): n in [1..30]]; // _Vincenzo Librandi_, Nov 29 2016
%Y Cf. A001003, A005061, A001263, A126216, A269730, A269732.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_, Mar 08 2016
%E More terms from _Gheorghe Coserea_, Apr 23 2016
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