A121545 Coefficients of Taylor series expansion of the operad Prim L.
0, 1, 1, 4, 17, 81, 412, 2192, 12049, 67891, 390041, 2276176, 13455356, 80402284, 484865032, 2947107384, 18036248337, 111046920567, 687345582787, 4274642610932, 26697307240777, 167377288848977
Offset: 0
Links
- Olivier Gérard and Vincenzo Librandi, Table of n, a(n) for n = 0..200 (first 51 terms from Olivier Gérard)
- Francesca Aicardi, Fuss-Catalan Triangles, arXiv:2310.07317 [math.CO], 2023.
- Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
- Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
- Philippe Leroux, An equivalence of categories motivated by weighted directed graphs, arXiv:math-ph/0709.3453, 2007-2008.
Programs
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Maple
a := n -> ifelse(n = 0, 0, binomial(3*n - 2, n - 1)*hypergeom([2, 1 - n], [2 - 3*n], -1) / n): seq(simplify(a(n)), n = 0..21); # Peter Luschny, Oct 09 2022
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Mathematica
CoefficientList[Series[Sin[1/3*ArcSin[Sqrt[27*x/4]]]^2/(3/4 + Sin[1/3*ArcSin[Sqrt[27*x/4]]]^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 19 2012 *) -
Maxima
a(n):=sum(k*(-1)^(k+1)*binomial(3*n-k-1,n-k),k,1,n)/n; /* Vladimir Kruchinin, Oct 09 2022 */
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PARI
{a(n)=local(G=1); if(n<1,0,for(i=1,n,G=1+x*G^3+O(x^(n+1))); polcoeff(x*G^2/(1+x*G^2),n))} \\ Paul D. Hanna, Nov 03 2012 -
PARI
x='x+O('x^22); concat(0, Vec(serreverse(x*(2*x-1)^2/(1-x)^3))) \\ Gheorghe Coserea, Aug 18 2017
Formula
G.f.: sin^2( (1/3)*arcsin(sqrt(27*x/4)) ) / ( 3/4 + sin^2( (1/3)*arcsin(sqrt(27*x/4)) )).
G.f.: x*G(x)^2 / (1 + x*G(x)^2), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Nov 03 2012
From Gary W. Adamson, Jul 13 2011: (Start)
As to a signed variant for n > 0: (1, -1, 4, -17, ...), a(n) = upper left term of M^n, M = the following infinite square production matrix:
1, 1, 0, 0, 0, 0, ...
-2, -2, 1, 0, 0, 0, ...
3, 3, -2, 1, 0, 0, ...
-4, -4, 3, -2, 1, 0, ...
5, 5, -4, 3, -2, 1, ...
-6, -6, 5, -4, 3, -2, ...
...
(each column is (1, -2, 3, -4, 5, ...) prepended with (0, 0, 1, 2, 3, ...) zeros by columns). (End)
Recurrence: 32*n*(2*n-1)*a(n) = 16*(11*n^2 - n - 15)*a(n-1) + 6*(278*n^2 - 1351*n + 1670)*a(n-2) + 45*(3*n-8)*(3*n-7)*a(n-3). - Vaclav Kotesovec, Nov 19 2012
a(n) ~ 3^(3*n+1/2)/(2^(2*n+4)*n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Nov 19 2012
From Peter Bala, Feb 04 2022: (Start)
G.f. A(x) = (G(x) - 1)/(2*G(x) - 1), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
Let B(x) = 2 + x + 2*x^2 + 6*x^3 + 22*x^4 + 91*x^5 + ... denote the o.g.f. of A000139. Then A(x) = x*C(x)'/C(x), where C(x) = 1 + x*(B(x) - 1).
Equivalently, exp(Sum_{n >= 1} a(n)*x^n/n) = C(x), a power series with integer coefficients. It follows that the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all prime p and positive integers n and k. (End)
a(n) = (1/n)*Sum_{k=1..n} k*(-1)^(k+1)*C(3*n-k-1,n-k). - Vladimir Kruchinin, Oct 09 2022
a(n) = binomial(3*n-2, n-1)*hypergeom([2, 1-n], [2-3*n], -1) / n for n >= 1. - Peter Luschny, Oct 09 2022
Extensions
More terms from Olivier Gérard, Oct 11 2007