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A269732
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Dimensions of the 4-polytridendriform operad TDendr_4.
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4
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1, 9, 101, 1269, 17081, 240849, 3511741, 52515549, 801029681, 12414177369, 194922521301, 3094216933509, 49575333021801, 800645021406369, 13020241953611181, 213025792632813549, 3504075376813414241, 57914491106005287849, 961297812844696640581, 16017765308027639317269, 267831397282643166904601, 4492625888792276208945009, 75578709400747348254905501
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = P_n(4), 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) = 9*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Apr 24 2016
a(n) ~ sqrt(40 + 18*sqrt(5)) * (9 + 4*sqrt(5))^n / (40*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 24 2016
a(n) ~ phi^(6*n + 3) / (2^(5/2) * 5^(3/4) * sqrt(Pi) * n^(3/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 23 2017
O.g.f.: A(x) = (1 - sqrt(1 - 18*x + x^2) - 9*x)/(40*x). - Peter Bala, Jan 25 2018
a(n) = (1/(2*m*(m+1))) * Integral_{x = 1..2*m+1} Legendre_P(n,x) dx at m = 4.
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 = 4. (End)
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MATHEMATICA
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Rest[CoefficientList[Series[(1 - 9*x - Sqrt[1 - 18*x + x^2])/(40*x), {x, 0, 20}], x]] (* Vaclav Kotesovec, Apr 24 2016 *)
Table[-I*LegendreP[n, -1, 2, 9]/(2*Sqrt[5]), {n, 1, 20}] (* Vaclav Kotesovec, Apr 24 2016 *)
RecurrenceTable[{a[1] == 1, a[2] == 9, (n+1) a[n] == 9 (2 n - 1) a[n-1] - (n - 2) a[n-2]}, a, {n, 25}] (* Vincenzo Librandi, Nov 29 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));
(PARI) my(q=4, x='x + O('x^24)); Vec(serreverse(x/((1+q*x)*(1+(q+1)*x)))) \\ Gheorghe Coserea, Sep 30 2017
(Magma) I:=[1, 9]; [n le 2 select I[n] else (9*(2*n-1)*Self(n-1)-(n-2)*Self(n-2))/(n+1): n in [1..30]]; // Vincenzo Librandi, Nov 29 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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