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.

A328006 Expansion of e.g.f. 1 / (1 - Sum_{k>=1} binomial(2*k,k) * x^k / (k + 1)!).

Original entry on oeis.org

1, 1, 4, 23, 174, 1642, 18596, 245737, 3711294, 63056858, 1190408544, 24720216578, 560011664724, 13743710272060, 363241612472368, 10286092411744025, 310694791014710206, 9971177817032175594, 338830529059491098336, 12153453467291303419246, 458873804279349884222364
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 01 2019

Keywords

Crossrefs

Programs

  • Maple
    seq(n!*coeff(series(1/(2 - exp(2*x) * (BesselI(0, 2*x) - BesselI(1, 2*x))), x, 21), x, n), n = 0..20); # Vaclav Kotesovec, Oct 02 2019
  • Mathematica
    nmax = 20; CoefficientList[Series[1/(2 - Exp[2 x] (BesselI[0, 2 x] - BesselI[1, 2 x])), {x, 0, nmax}], x] Range[0, nmax]!
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] CatalanNumber[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Formula

E.g.f.: 1 / (2 - exp(2*x) * (BesselI(0,2*x) - BesselI(1,2*x))).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A000108(k) * a(n-k).
a(n) ~ n! / (exp(2*r)*(BesselI(0, 2*r) - BesselI(2, 2*r)) * r^(n+1)), where r = 0.52970787846036422338310218180536596363570735225100094676866... is the root of the equation exp(2*r)*(BesselI(0,2*r) - BesselI(1,2*r)) = 2. - Vaclav Kotesovec, Oct 02 2019

A330019 Expansion of e.g.f. 1 / (1 - Sum_{k>=1} binomial(k,floor(k/2)) * x^k / k!).

Original entry on oeis.org

1, 1, 4, 21, 150, 1330, 14180, 176295, 2505230, 40049226, 711379872, 13899553206, 296270826852, 6841305568812, 170127212242416, 4532854743105975, 128824523061126750, 3890041395675793930, 124375112406132404960, 4197530354920789582410, 149118181703716510545260
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 27 2019

Keywords

Crossrefs

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[1/(2 - BesselI[0, 2 x] - BesselI[1, 2 x]), {x, 0, nmax}], x] Range[0, nmax]!
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k, Floor[k/2]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Formula

E.g.f.: 1 / (2 - BesselI(0,2*x) - BesselI(1,2*x)).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A001405(k) * a(n-k).
Showing 1-2 of 2 results.