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A269730
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Dimensions of the 2-polytridendriform operad TDendr_2.
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3
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1, 5, 31, 215, 1597, 12425, 99955, 824675, 6939769, 59334605, 513972967, 4501041935, 39784038517, 354455513105, 3179928556219, 28701561707675, 260447708523505, 2374690737067925, 21744508765633327, 199877846477679815, 1843718766426242221, 17060955558786455705, 158333204443000060291
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OFFSET
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1,2
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LINKS
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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.
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A001047, A001263, A126216, A269731, A269732, A001003, A006442.
Sequence in context: A153292 A087457 A146962 * A036758 A153232 A287899
Adjacent sequences: A269727 A269728 A269729 * A269731 A269732 A269733
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Mar 08 2016
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EXTENSIONS
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More terms from Gheorghe Coserea, Apr 23 2016
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STATUS
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approved
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