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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 19 2006
EXTENSIONS
New name from Paul D. Hanna, May 30 2015
STATUS
approved