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.

A308847 Expansion of e.g.f. exp(-2*x) / BesselI(0,2*x).

Original entry on oeis.org

1, -2, 2, 4, -14, -52, 284, 1496, -10958, -74372, 681652, 5656616, -62226116, -610306712, 7832965352, 88645228304, -1300254163918, -16676932459172, 275196007522436, 3944890321174664, -72330003541955564, -1145979961718846152, 23112838345877865752, 401070175407076000624, -8824400094691629670724
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 28 2019

Keywords

Comments

E.g.f. is inverse of e.g.f. for A000984 (central binomial coefficients).

Crossrefs

Cf. A000984, A002420, A178955, A308848.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[Exp[-2 x]/BesselI[0, 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n, k] Binomial[2 k, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(-2*x) / besseli(0,2*x))) \\ Michel Marcus, Jul 02 2019

Formula

E.g.f.: 1 / Sum_{k>=0} binomial(2*k,k)*x^k/k!.

A178956 Numerators in expansion of 1/(Sum_{n >= 0} (x^n/n!)*binomial(2n,n)/(n+1)).

Original entry on oeis.org

1, -1, 0, 1, 1, -1, -7, -13, 169, 233, 17, -18847, -43957, 26023, 429007, 31505483, -31381783, -753299101, -7372705283, 195152846389, 632396244331, -52258919107, -447065769153911, -13584264183406001, 34831349327448893, 674219621951483119, 2113978805943481381, -6602691591860999071, -370308752490094361609, -95867348590727618473, 752151788353653780085507
Offset: 0

Views

Author

N. J. A. Sloane, Dec 31 2010

Keywords

Comments

This is the reciprocal of the e.g.f. for the Catalan numbers.

Examples

			1, -1, 0, 1/6, 1/12, -1/60, -7/180, -13/1008, 169/20160, 233/25920, 17/14175, -18847/6652800, -43957/23950080, 26023/141523200, 429007/544864320, 31505483/100590336000, ...

Crossrefs

Cf. A000108, A178957 (denominators), A178955, A144186/A144187.

A178957 Denominators in expansion of 1/(Sum_{n >= 0} (x^n/n!)*binomial(2n,n)/(n+1)).

Original entry on oeis.org

1, 1, 1, 6, 12, 60, 180, 1008, 20160, 25920, 14175, 6652800, 23950080, 141523200, 544864320, 100590336000, 213497856000, 3952082534400, 200074178304000, 3577797070848000, 15595525693440000, 51711479930880000, 28100018194440192000, 1846572624206069760000, 14101100039391805440000, 168600109166641152000000, 2100476360034404352000000, 6405217323775501271040000, 418802671169936621568000000, 2506168365572477878272000000, 2368329105465991594967040000000
Offset: 0

Views

Author

N. J. A. Sloane, Dec 31 2010

Keywords

Comments

This is the reciprocal of the e.g.f. for the Catalan numbers.

Crossrefs

Cf. A000108, A178956 (numerators), A178955, A144186/A144187.

A308850 Expansion of e.g.f. exp(-2*x) / (BesselI(0,2*x) + BesselI(1,2*x)).

Original entry on oeis.org

1, -3, 8, -17, 18, 58, -364, 369, 6194, -37382, -28848, 1717274, -8592644, -47472804, 918146560, -2911313551, -61122074382, 806675821162, 46813084592, -105331573943466, 1018198168087636, 6417696715221572, -247555432672498872, 1535509971584425358, 34028097257000628028, -764203552200012087252
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 28 2019

Keywords

Comments

E.g.f. is inverse of e.g.f. for A001700.

Crossrefs

Cf. A001700, A178955, A246432, A308847, A308849.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[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] Binomial[2 k + 1, k + 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 25}]

Formula

E.g.f.: 1 / Sum_{k>=0} binomial(2*k+1,k+1)*x^k/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
Showing 1-5 of 5 results.

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

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