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%I #64 Oct 12 2023 07:46:53
%S 0,1,1,4,17,81,412,2192,12049,67891,390041,2276176,13455356,80402284,
%T 484865032,2947107384,18036248337,111046920567,687345582787,
%U 4274642610932,26697307240777,167377288848977
%N Coefficients of Taylor series expansion of the operad Prim L.
%H Olivier Gérard and Vincenzo Librandi, <a href="/A121545/b121545.txt">Table of n, a(n) for n = 0..200</a> (first 51 terms from Olivier Gérard)
%H Francesca Aicardi, <a href="https://arxiv.org/abs/2310.07317">Fuss-Catalan Triangles</a>, arXiv:2310.07317 [math.CO], 2023.
%H Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%H Isaac DeJager, Madeleine Naquin, and Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%H Philippe Leroux, <a href="http://arxiv.org/abs/0709.3453">An equivalence of categories motivated by weighted directed graphs</a>, arXiv:math-ph/0709.3453, 2007-2008.
%F G.f.: sin^2( (1/3)*arcsin(sqrt(27*x/4)) ) / ( 3/4 + sin^2( (1/3)*arcsin(sqrt(27*x/4)) )).
%F 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
%F From _Gary W. Adamson_, Jul 13 2011: (Start)
%F 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:
%F 1, 1, 0, 0, 0, 0, ...
%F -2, -2, 1, 0, 0, 0, ...
%F 3, 3, -2, 1, 0, 0, ...
%F -4, -4, 3, -2, 1, 0, ...
%F 5, 5, -4, 3, -2, 1, ...
%F -6, -6, 5, -4, 3, -2, ...
%F ...
%F (each column is (1, -2, 3, -4, 5, ...) prepended with (0, 0, 1, 2, 3, ...) zeros by columns). (End)
%F 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
%F a(n) ~ 3^(3*n+1/2)/(2^(2*n+4)*n^(3/2)*sqrt(Pi)). - _Vaclav Kotesovec_, Nov 19 2012
%F From _Peter Bala_, Feb 04 2022: (Start)
%F 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.
%F 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).
%F 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)
%F a(n) = (1/n)*Sum_{k=1..n} k*(-1)^(k+1)*C(3*n-k-1,n-k). - _Vladimir Kruchinin_, Oct 09 2022
%F 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
%p 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
%t 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 *)
%o (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
%o (PARI) x='x+O('x^22); concat(0, Vec(serreverse(x*(2*x-1)^2/(1-x)^3))) \\ _Gheorghe Coserea_, Aug 18 2017
%o (Maxima)
%o a(n):=sum(k*(-1)^(k+1)*binomial(3*n-k-1,n-k),k,1,n)/n; /* _Vladimir Kruchinin_, Oct 09 2022 */
%Y Cf. A000139, A006013, A001764.
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_, Oct 07 2007
%E More terms from _Olivier Gérard_, Oct 11 2007
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