A269732 Dimensions of the 4-polytridendriform operad TDendr_4.

1, 9, 101, 1269, 17081, 240849, 3511741, 52515549, 801029681, 12414177369, 194922521301, 3094216933509, 49575333021801, 800645021406369, 13020241953611181, 213025792632813549, 3504075376813414241, 57914491106005287849, 961297812844696640581, 16017765308027639317269, 267831397282643166904601, 4492625888792276208945009, 75578709400747348254905501

Offset: 1

Author

N. J. A. Sloane, Mar 08 2016

Keywords

• nonn
• easy

Links

• 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.

Crossrefs

Cf. A001003, A005060, A001263, A126216, A269730, A269731, A084769.

Programs

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

Mathematica

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 *)

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=4); vector(23, n, dimTDendr(n,q)) \\ Gheorghe Coserea, Apr 23 2016

PARI

my(q=4, x='x + O('x^24)); Vec(serreverse(x/((1+q*x)*(1+(q+1)*x)))) \\ Gheorghe Coserea, Sep 30 2017

Formula

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

A(x) = -serreverse(A005060(x))(-x). - Gheorghe Coserea, Sep 30 2017

O.g.f.: A(x) = (1 - sqrt(1 - 18*x + x^2) - 9*x)/(40*x). - Peter Bala, Jan 25 2018

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 = 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)

Extensions

More terms from Gheorghe Coserea, Apr 23 2016