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URL: https://oeis.org/A269731

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A269731
Dimensions of the 3-polytridendriform operad TDendr_3.
3
1, 7, 61, 595, 6217, 68047, 770149, 8939707, 105843409, 1273241431, 15517824973, 191202877411, 2377843390873, 29807864423071, 376255282112629, 4778240359795147, 61007205215610529, 782648075371992487, 10083436451634033757, 130413832663780730995, 1692599303723819234281, 22037570163808433691247, 287762084009227350367621
OFFSET
1,2
LINKS
Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016; Adv. Appl. Math., 77, 3-85, 2016.
FORMULA
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.
Recurrence: (n+1)*a(n) = 7*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Apr 24 2016
a(n) ~ sqrt(24 + 14*sqrt(3)) * (7 + 4*sqrt(3))^n / (24*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 24 2016
A(x) = -serreverse(A005061(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 = 3.
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)
MATHEMATICA
Rest[CoefficientList[Series[(1 - 7*x - Sqrt[1 - 14*x + x^2])/(24*x), {x, 0, 20}], x]] (* Vaclav Kotesovec, Apr 24 2016 *)
Table[-I*LegendreP[n, -1, 2, 7]/(2*Sqrt[3]), {n, 1, 20}] (* Vaclav Kotesovec, Apr 24 2016 *)
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 *)
PROG
(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=3); vector(23, n, dimTDendr(n, q)) \\ Gheorghe Coserea, Apr 23 2016
(PARI) my(q=3, x='x + O('x^24)); Vec(serreverse(x/((1+q*x)*(1+(q+1)*x)))) \\ Gheorghe Coserea, Sep 30 2017
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Mar 08 2016
EXTENSIONS
More terms from Gheorghe Coserea, Apr 23 2016
STATUS
approved