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A121545
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Coefficients of Taylor series expansion of the operad Prim L.
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2
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0, 1, 1, 4, 17, 81, 412, 2192, 12049, 67891, 390041, 2276176, 13455356, 80402284, 484865032, 2947107384, 18036248337, 111046920567, 687345582787, 4274642610932, 26697307240777, 167377288848977
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OFFSET
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0,4
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LINKS
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FORMULA
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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
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
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) = binomial(3*n-2, n-1)*hypergeom([2, 1-n], [2-3*n], -1) / n for n >= 1. - Peter Luschny, Oct 09 2022
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MAPLE
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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
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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 *)
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PROG
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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
(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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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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