cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A216316 G.f.: 1/( (1-8*x)*(1+x)^2 )^(1/3).

Original entry on oeis.org

1, 2, 13, 80, 538, 3740, 26650, 193160, 1417945, 10511450, 78533629, 590485208, 4463274232, 33886781840, 258260802232, 1974759985952, 15143163422794, 116417053435316, 896996316176290, 6925241271855296, 53562550587963052, 414948608904171464, 3219356873886333676
Offset: 0

Views

Author

Paul D. Hanna, Sep 03 2012

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 13*x^2 + 80*x^3 + 538*x^4 + 3740*x^5 + 26650*x^6 +...
where 1/A(x)^3 = 1 - 6*x - 15*x^2 - 8*x^3.
The logarithm of the g.f. begins:
log(A(x)) = 2*x + 22*x^2/2 + 170*x^3/3 + 1366*x^4/4 + 10922*x^5/5 + 87382*x^6/6 +...+ A007613(n)*x^n/n +...

Links

Crossrefs

Cf. A004987, A007613, A216317, A216357, A216358.

Programs

  • Mathematica
    CoefficientList[Series[1/((1-8*x)*(1+x)^2)^(1/3), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
  • PARI
    {a(n)=polcoeff(1/( (1-8*x)*(1+x)^2 +x*O(x^n) )^(1/3),n)}
  • PARI
    {a(n)=local(A=1+x); A=exp(sum(m=1, n+1, sum(j=0, m, binomial(3*m, 3*j))*x^m/m +x*O(x^n))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} A007613(n)*x^n/n ), where A007613(n) = Sum_{k=0..n} binomial(3*n,3*k).
Recurrence: n*a(n) = (7*n-5)*a(n-1) + 8*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ Gamma(2/3)*2^(3*n+1)/(3^(5/6)*Pi*n^(2/3)). - Vaclav Kotesovec, Oct 20 2012
Inverse binomial transform of A004987. - Peter Bala, Jul 02 2023

A216358 G.f.: 1/( (1-32*x)*(1+11*x-x^2)^2 )^(1/5).

Original entry on oeis.org

1, 2, 129, 2258, 66266, 1711282, 48405689, 1366932878, 39516211006, 1152710434262, 33978897474149, 1008971023405798, 30155867955237721, 906105094582017192, 27351768342997448884, 828919276503075367768, 25208280600556937464286, 768948732346237772809572
Offset: 0

Views

Author

Paul D. Hanna, Sep 04 2012

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 129*x^2 + 2258*x^3 + 66266*x^4 + 1711282*x^5 +...
where 1/A(x)^5 = 1 - 10*x - 585*x^2 - 3830*x^3 + 705*x^4 - 32*x^5.
The logarithm of the g.f. begins:
log(A(x)) = 2*x + 254*x^2/2 + 6008*x^3/3 + 215766*x^4/4 + 6643782*x^5/5 + 215492564*x^6/6 +...+ A070782(n)*x^n/n +...

Crossrefs

Cf. A216316, A216357, A070782.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(5*m, 5*j))*x^m/m+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} A070782(n)*x^n/n ) where A070782(n) = Sum_{k=0..n} binomial(5*n,5*k).
a(n) ~ 2^(5*n+3) * ((25-11*sqrt(5))/2)^(1/10) * GAMMA(4/5) / (5 * 11^(2/5) * n^(4/5) * Pi). - Vaclav Kotesovec, Jul 31 2014
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.