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 1-5 of 5 results.

A323666 Expansion of e.g.f. exp(exp(2*x)*BesselI(0,2*x) - 1).

Original entry on oeis.org

1, 2, 10, 64, 498, 4544, 47272, 549448, 7032338, 98034816, 1475781592, 23821854808, 409932257560, 7483462406840, 144320890075608, 2929683071286416, 62402858556637970, 1390821290318306688, 32355429437927804952, 783919832399050511928, 19741529222451177258920, 515813862624032150918280
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 23 2019

Keywords

Crossrefs

Cf. A000984, A304788.

Programs

  • Maple
    seq(n!*coeff(series(exp(exp(2*x)*BesselI(0,2*x)-1),x=0,22),x,n),n=0..21); # Paolo P. Lava, Jan 28 2019
  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[Exp[2 x] BesselI[0, 2 x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Binomial[2 k, k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]
  • PARI
    my(x='x + O('x^25)); Vec(serlaplace(exp(exp(2*x)*besseli(0, 2*x)-1))) \\ Michel Marcus, Jan 24 2019

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A000984(k)*binomial(n-1,k-1)*a(n-k).

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

Cf. A000108, A001700, A088218, A178955, A304788, A328004.

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

A323667 Expansion of e.g.f. exp(BesselI(0,2*x) + BesselI(1,2*x) - 1).

Original entry on oeis.org

1, 1, 3, 10, 43, 211, 1191, 7463, 51535, 386809, 3133273, 27184620, 251253157, 2461527511, 25459020289, 276987375642, 3160197122183, 37705878268985, 469340324930493, 6081394853597162, 81866045488063721, 1142928276326927223, 16521454311961005245, 246917508673451732077
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 23 2019

Keywords

Crossrefs

Cf. A001405, A302197, A304788.

Programs

  • Maple
    seq(n!*coeff(series(exp(BesselI(0,2*x)+BesselI(1,2*x)-1),x=0,24),x,n),n=0..23); # Paolo P. Lava, Jan 28 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[BesselI[0, 2 x] + BesselI[1, 2 x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Binomial[k, Floor[k/2]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
  • PARI
    my(x='x + O('x^25)); Vec(serlaplace(exp(besseli(0, 2*x)+x*besseli(1, 2*x)-1))) \\ Michel Marcus, Jan 24 2019

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A001405(k)*binomial(n-1,k-1)*a(n-k).

A323668 Expansion of e.g.f. exp(exp(2*x)*(BesselI(0,2*x) + BesselI(1,2*x)) - 1).

Original entry on oeis.org

1, 3, 19, 152, 1467, 16445, 208471, 2934321, 45254447, 756995131, 13623709401, 262067291106, 5358900661509, 115953603121881, 2644399031839729, 63346390393538780, 1589177904965680263, 41642328796769014811, 1137083068108603968349, 32287430515011314674632, 951565685429585731747913
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 23 2019

Keywords

Crossrefs

Cf. A001700, A302197, A304788.

Programs

  • Maple
    seq(n!*coeff(series(exp(exp(2*x)*(BesselI(0,2*x)+BesselI(1,2*x))-1),x=0,21),x,n),n=0..20); # Paolo P. Lava, Jan 28 2019
  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[Exp[2 x] (BesselI[0, 2 x] + BesselI[1, 2 x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Binomial[2 k + 1, k + 1] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}]
  • PARI
    my(x='x + O('x^25)); Vec(serlaplace(exp(exp(2*x)*(besseli(0, 2*x)+x*besseli(1, 2*x))-1))) \\ Michel Marcus, Jan 24 2019

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A001700(k)*binomial(n-1,k-1)*a(n-k).

A323672 Expansion of e.g.f. exp(exp(x)*BesselI(1,2*x)/x - 1).

Original entry on oeis.org

1, 1, 3, 11, 50, 267, 1633, 11195, 84745, 700332, 6262087, 60146704, 616841371, 6720592647, 77453291594, 940701503518, 12001369992614, 160373548837863, 2238995849944224, 32584082576638311, 493289434851850445, 7754247825363510168, 126354223534523911670, 2131014061115010861375
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 23 2019

Keywords

Crossrefs

Cf. A001006, A304788.

Programs

  • Maple
    seq(n!*coeff(series(exp(exp(x)*BesselI(1,2*x)/x-1),x=0,24),x,n), n=0..23); # Paolo P. Lava, Jan 28 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[Exp[x] BesselI[1, 2 x]/x - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Hypergeometric2F1[(1 - k)/2, -k/2, 2, 4] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A001006(k)*binomial(n-1,k-1)*a(n-k).
Showing 1-5 of 5 results.

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