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

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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

A216357 Expansion of 1/( (1-16*x)*(1+4*x)^2 )^(1/4).

Original entry on oeis.org

1, 2, 38, 404, 5510, 74492, 1048924, 15004776, 217943238, 3200089580, 47405806708, 707305846936, 10616181542044, 160142807848792, 2426097698458360, 36890818642990544, 562772826273060678, 8609639617006367052, 132048790603779592196, 2029851945081220214200
Offset: 0

Views

Author

Paul D. Hanna, Sep 04 2012

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 38*x^2 + 404*x^3 + 5510*x^4 + 74492*x^5 + 1048924*x^6 + ...
where 1/A(x)^4 = 1 - 8*x - 112*x^2 - 256*x^3.
The logarithm of the g.f. begins:
log(A(x)) = x + 2*x^2/2 + 72*x^3/3 + 992*x^4/4 + 16512*x^5/5 + 261632*x^6/6 + 4196352*x^7/7 + ... + A070775(n)*x^n/n + ...

Links

Crossrefs

Cf. A070775, A216316, A216358.

Programs

  • Maple
    f:= gfun:-rectoproc({(48+64*n)*a(n)+(14+12*n)*a(n+1)+(-n-2)*a(n+2), a(0) = 1, a(1) = 2}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Dec 09 2016
  • Mathematica
    a = DifferenceRoot[Function[{a, n}, {(48+64n) a[n] + (14+12n) a[1+n] + (-2-n) a[2+n] == 0, a[0] == 1, a[1] == 2}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Sep 16 2022, after Robert Israel *)
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(4*m, 4*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} A070775(n)*x^n/n) where A070775(n) = Sum_{k=0..n} binomial(4*n,4*k).
a(n) ~ GAMMA(3/4) * 2^(4*n+1/2) / (Pi* sqrt(5) * n^(3/4)). - Vaclav Kotesovec, Jul 31 2014
a(n) = ((64*n-80)*a(n-2)+(12*n-10)*a(n-1))/n. - Robert Israel, Dec 09 2016
Showing 1-2 of 2 results.

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