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.

Previous Showing 11-14 of 14 results.

A292631 a(n) = n! * [x^n] exp(n*x)*(BesselI(0,2*x) + BesselI(1,2*x)).

Original entry on oeis.org

1, 2, 10, 75, 758, 9660, 148772, 2688420, 55784710, 1307378358, 34158527852, 984547901051, 31034429035260, 1062081192039140, 39218355263626632, 1554260970293874135, 65803396940022289734, 2964120950479432183950, 141548149894016562758300, 7143010414313948156920665, 379821534884560034711455956
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 20 2017

Keywords

Comments

The n-th term of the n-th binomial transform of A001405.

Links

Crossrefs

Main diagonal of A292630.
Cf. A001405, A186925, A292629, A292632.

Programs

  • Mathematica
    Table[n!*SeriesCoefficient[E^(n*x)*(BesselI[0,2*x] + BesselI[1,2*x]),{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 20 2017 *)

Formula

a(n) = A292630(n,n).
a(n) ~ (BesselI(0,2) + BesselI(1,2)) * n^n. - Vaclav Kotesovec, Sep 20 2017

A307844 Constant term in the expansion of (n/x + 1 + n*x)^n.

Original entry on oeis.org

1, 1, 9, 55, 1729, 19251, 1050841, 16977129, 1322929665, 28017221059, 2839212609001, 74390784295653, 9283240524317761, 289865990675075725, 42976734096778661817, 1557837326400792009751, 267561369300137776050177, 11042876765198762014337235
Offset: 0

Views

Author

Seiichi Manyama, May 01 2019

Keywords

Comments

Also coefficient of x^n in the expansion of (1 + x + (n*x)^2)^n.
Also coefficient of x^n in the expansion of 1/sqrt(1 - 2*x + (1-4*n^2)*x^2).

Links

Crossrefs

Main diagonal of A307847.
Cf. A002426, A092366, A186925, A187018.

Programs

  • Mathematica
    Flatten[{1, Table[Sum[(-1)^k * (2*n + 1)^(n-k) * n^k * Binomial[n,k] * Binomial[2*k,k], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 02 2019 *)
  • PARI
    {a(n) = polcoef((n/x+1+n*x)^n, 0)}
  • PARI
    {a(n) = polcoef((1+x+(n*x)^2)^n, n)}
  • PARI
    {a(n) = sum(k=0, n, (1-2*n)^(n-k)*n^k*binomial(n, k)*binomial(2*k, k))}
  • PARI
    {a(n) = sum(k=0, n, (1+2*n)^(n-k)*(-n)^k*binomial(n, k)*binomial(2*k, k))}
  • PARI
    {a(n) = sum(k=0, n\2, n^(2*k)*binomial(n, 2*k)*binomial(2*k, k))}

Formula

a(n) = Sum_{k=0..n} (1-2*n)^(n-k) * n^k * binomial(n,k) * binomial(2*k,k).
a(n) = Sum_{k=0..n} (1+2*n)^(n-k) * (-n)^k * binomial(n,k) * binomial(2*k,k).
a(n) = Sum_{k=0..floor(n/2)} n^(2*k) *binomial(n,2*k) * binomial(2*k,k).
a(n) ~ (exp(1/2) + (-1)^n * exp(-1/2)) * 2^(n - 1/2) * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, May 02 2019

A293491 a(n) = n! * [x^n] exp((n+2)*x)*BesselI(0,2*x).

Original entry on oeis.org

1, 3, 18, 155, 1734, 23877, 390804, 7417377, 160256070, 3885021569, 104465601756, 3086353547433, 99399100528924, 3466411543407555, 130151205663179112, 5235127829223881895, 224609180728848273990, 10239557195235638377449, 494317596005491398892620, 25192788307121307053168673
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 10 2017

Keywords

Comments

The n-th term of the n-th binomial transform of A000984.

Links

Crossrefs

Cf. A000984, A026375, A081671, A098409, A098410, A104454, A186925, A292627, A292632.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[(n + 2) x] BesselI[0, 2 x], {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[1/Sqrt[(1 - n x) (1 - (n + 4) x)], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[Sum[Binomial[n, k] Binomial[2 k, k] n^(n - k), {k, 0, n}], {n, 1, 19}]]
Table[(n + 2)^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {1}, 4/(2 + n)^2], {n, 0, 19}]

Formula

a(n) = [x^n] 1/sqrt((1 - n*x)*(1 - (n + 4)*x)).
a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k)*n^(n-k).
a(n) ~ exp(2) * BesselI(0,2) * n^n. - Vaclav Kotesovec, Oct 16 2017

A294409 a(n) = n! * [x^n] exp(n*x)*BesselI(0,2*n*x).

Original entry on oeis.org

1, 1, 12, 189, 4864, 159375, 6578496, 323652399, 18572378112, 1216112914971, 89530000000000, 7319100286183983, 657910135976361984, 64494528072860946073, 6847518630093139525632, 782782183702056884765625, 95860848315529046085599232, 12520224284071636768582166787, 1737254440584625641929018966016
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 30 2017

Keywords

Comments

a(n) is the central coefficient of (1 + n*x + n^2*x^2)^n.

Links

Crossrefs

Cf. A000312, A002426, A098453, A186925, A292629.

Programs

  • Maple
    seq(coeff((1+n*x+n^2*x^2)^n,x,n),n=0..100); # Robert Israel, Oct 30 2017
  • Mathematica
    Table[n! SeriesCoefficient[Exp[n x] BesselI[0, 2 n x], {x, 0, n}], {n, 0, 18}]
Table[CoefficientList[Series[(1 + n x + n^2 x^2)^n, {x, 0, n}], x][[-1]], {n, 0, 18}]
Table[SeriesCoefficient[1/Sqrt[(1 + n x) (1 - 3 n x)], {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[n^n Sum[Binomial[n, k] Binomial[k, n - k],{k, 0, n}], {n, 1, 18}]]
Join[{1}, Table[n^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {1}, 4], {n, 1, 18}]]

Formula

a(n) = [x^n] 1/sqrt((1 + n*x)*(1 - 3*n*x)).
a(n) = A000312(n)*A002426(n).
a(n) ~ sqrt(3)*3^n*n^n/(2*sqrt(Pi*n)).
Previous Showing 11-14 of 14 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.