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

0,3

Comments

Previous name was: Column 0 and row sums of symmetric triangle A126150.

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.

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

Example

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)).

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

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 *)

Prog

(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), ", "))
(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), ", "))

Crossrefs

Cf. A126150; diagonals: A126152, A126153.

Cf. A000326.

Sequence in context: A346184 A304646 A251576 * A066319 A186269 A111826

Adjacent sequences: A126148 A126149 A126150 * A126152 A126153 A126154

Keyword

nonn

Author

Paul D. Hanna, Dec 19 2006

Extensions

New name from Paul D. Hanna, May 30 2015

Status

approved