A126151 E.g.f.: ( (1 + cos(sqrt(6)*x))/2 )^(-1/3), showing coefficients of only the even powers of x.
1, 1, 6, 96, 2976, 151416, 11449296, 1204566336, 168233625216, 30110372009856, 6719377991060736, 1829013279998846976, 596449130341224185856, 229556544889929225117696, 102956750031135241952280576, 53228316147100497514507862016, 31423560379886826670772937424896
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + x^2/2! + 6*x^4/4! + 96*x^6/6! + 2976*x^8/8! + 151416*x^10/10! +... where the logarithm begins: log(A(x)) = x^2/2! + 3*x^4/4! + 36*x^6/6! + 918*x^8/8! + 40176*x^10/10! + 2686608*x^12/12! +... compare the logarithm to A(x)^3 = 1 + 3*x^2/! + 36*x^4/4! + 918*x^6/6! + 40176*x^8/8! + 2686608*x^10/10! +... where A(x)^3 = 2/(1 + cos(sqrt(6)*x)).
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..232
- E. Norton, Symplectic Reflection Algebras in Positive Characteristic as Ore Extensions, arXiv preprint arXiv:1302.5411 [math.RA], 2013.
Programs
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Maple
A000326 := n -> n * (3 * n - 1) / 2; T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1) else A000326(n - k + 1) * T(n, k - 1) + T(n - 1, k) fi fi end: a := n -> T(n, n): seq(a(n), n = 0..16); # Peter Luschny, Sep 30 2023
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Mathematica
terms = 18; CoefficientList[((1 + Cos[Sqrt[6] x])/2)^(-3^(-1)) + O[x]^(2 terms), x] Range[0, 2 terms - 2]! // DeleteCases[#, 0]& (* Jean-François Alcover, Jul 26 2018 *)
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PARI
/* Continued Fraction involving pentagonal numbers A000326: */ {a(n)=local(CF=1+x*O(x),m,P); for(k=1, n,m=n-k+1;P=m*(3*m-1)/2; CF=1/(1-P*x*CF)); polcoeff(CF, n, x)} for(n=0,20,print1(a(n),","))
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PARI
/* E.g.f. A(x) = exp( Integral^2 A(x)^3 dx^2 ): */ {a(n)=local(A=1+x*O(x)); for(i=1, n, A=exp(intformal(intformal(A^3 + x*O(x^(2*n))))) ); (2*n)!*polcoeff(A, 2*n, x)} for(n=0,20,print1(a(n),", ")) -
PARI
/* E.g.f. A(x) = exp( Integral A(x)^(3/2) * Integral 1/A(x)^(3/2) dx dx ) */ {a(n) = local(A=1+x); for(i=1,n, A = exp( intformal( A^(3/2) * intformal( 1/A^(3/2) + x*O(x^n)) ) ) ); n!*polcoeff(A,n)} for(n=0,20,print1(a(2*n),", "))
Formula
a(n) = Sum_{0<=k<=n} A087736(n,k)*2^(n-k). - Philippe Deléham, Jul 17 2007
G.f.: 1/(1-x/(1-5*x/(1-12*x/(1-22*x/(1-35*x/(1-51*x/(1-70*x/(1-...- (n*(3*n-1)/2)*x/(1-...))))))))), a continued fraction involving pentagonal numbers A000326. - Paul D. Hanna, Feb 15 2012
E.g.f. satisfies: A(x) = exp( Integral Integral A(x)^3 dx dx ), where A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! and the constant of integration is zero. - Paul D. Hanna, May 29 2015
E.g.f. satisfies: A(x) = exp( Integral A(x)^(3/2) * Integral 1/A(x)^(3/2) dx dx ), where A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! and the constant of integration is zero. - Paul D. Hanna, Jun 02 2015
a(n) ~ Gamma(1/3) * 2^(3*n+4/3) * 3^(n+1/2) * n^(2*n+1/6) / (exp(2*n) * Pi^(2*n+7/6)). - Vaclav Kotesovec, May 30 2015
The computation can be based on the pentagonal numbers, a(n) = T(n, n) where T(n, k) = A000326(n - k + 1) * T(n, k - 1) + T(n - 1, k) for 0 < k < n, and T(n, 0) = 1, T(n, n) = T(n, n-1) if n > 0. This is equivalent to Paul D. Hanna's continued fraction 2012. - Peter Luschny, Sep 30 2023
Extensions
New name from Paul D. Hanna, May 30 2015
Comments