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 1-4 of 4 results.

A186925 Coefficient of x^n in (1+n*x+x^2)^n.

Original entry on oeis.org

1, 1, 6, 45, 454, 5775, 88796, 1602447, 33213510, 777665691, 20302315252, 584774029983, 18422140045596, 630132567760345, 23257790717110392, 921362075184792825, 38994274473840538182, 1755943506127367745795, 83829045032101462204100, 4229207755493569286374167
Offset: 0

Views

Author

Emanuele Munarini, Mar 02 2011

Keywords

Links

Crossrefs

Main diagonal of A292627.
Cf. A092366, A187018, A187019, A187021, A070910, A292629.

Programs

  • Magma
    P:=PolynomialRing(Integers()); [ Coefficients((1+n*x+x^2)^n)[n+1]: n in [0..22] ]; // Klaus Brockhaus, Mar 02 2011
  • Mathematica
    Flatten[{1,Table[Sum[Binomial[n, k]*Binomial[n-k, n-2*k]*n^(n-2*k), {k,0,Floor[n/2]}],{n,1,20}]}] (* Vaclav Kotesovec, Apr 17 2014 *)
Table[GegenbauerC[n, -n, -n/2] + KroneckerDelta[n, 0], {n, 0, 100}] (* Emanuele Munarini, Oct 20 2016 *)
Table[SeriesCoefficient[(1 + n*x + x^2)^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 13 2023 *)
  • Maxima
    a(n):=coeff(expand((1+n*x+x^2)^n),x,n);
  • Maxima
    makelist(ultraspherical(n,-n,-n/2),n,0,12); /* Emanuele Munarini, Oct 20 2016 */
makelist(a(n),n,0,20);
  • PARI
    {a(n) = sum(k=0, n, (n-2)^(n-k)*binomial(n, k)*binomial(2*k, k))} \\ Seiichi Manyama, May 01 2019
  • PARI
    a(n) = polcoef((1+n*x+x^2)^n, n); \\ Michel Marcus, May 01 2019

Formula

a(n) = [x^n] (1+n*x+x^2)^n.
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, n-2*k)*n^(n-2*k).
a(n) ~ BesselI(0,2) * n^n. - Vaclav Kotesovec, Apr 17 2014
a(n) = GegenbauerPoly(n,-n,-n/2). - Emanuele Munarini, Oct 20 2016
From Ilya Gutkovskiy, Sep 20 2017: (Start)
a(n) = [x^n] 1/sqrt((1 + 2*x - n*x)*(1 - 2*x - n*x)).
a(n) = n! * [x^n] exp(n*x)*BesselI(0,2*x). (End)
From Seiichi Manyama, May 01 2019: (Start)
a(n) = Sum_{k=0..n} (n-2)^(n-k) * binomial(n,k) * binomial(2*k,k).
a(n) = Sum_{k=0..n} (n+2)^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). (End)
a(n) = (1/4)^n * Sum_{k=0..n} (n-2)^k * (n+2)^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

A292628 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)*BesselI(1,2*x).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 2, 3, 0, 1, 4, 6, 0, 0, 1, 6, 15, 16, 10, 0, 1, 8, 30, 56, 45, 0, 0, 1, 10, 51, 144, 210, 126, 35, 0, 1, 12, 78, 304, 685, 792, 357, 0, 0, 1, 14, 111, 560, 1770, 3258, 3003, 1016, 126, 0, 1, 16, 150, 936, 3885, 10224, 15533, 11440, 2907, 0, 0, 1, 18, 195, 1456, 7570, 26550, 58947, 74280, 43758, 8350, 462
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 20 2017

Keywords

Comments

A(n,k) is the k-th binomial transform of A138364 evaluated at n.

Examples

			E.g.f. of column k: A_k(x) = x/1! + 2*k*x^2/2! + 3*(k^2 + 1)*x^3/3! + 4*k*(k^2 + 3)*x^4/4! + 5*(k^4 + 6*k^2 + 2)*x^5/5! + ...
Square array begins:
   0,   0,    0,    0,     0,     0,  ...
   1,   1,    1,    1,     1,     1,  ...
   0,   2,    4,    6,     8,    10,  ...
   3,   6,   15,   30,    51,    78,  ...
   0,  16,   56,  144,   304,   560,  ...
  10,  45,  210,  685,  1770,  3885,  ...

Links

Crossrefs

Columns k=0..3 give A138364, A005717, A001791, A026376.
Main diagonal gives A292629.
Cf. A292627.

Programs

  • Mathematica
    Table[Function[k, n! SeriesCoefficient[Exp[k x] BesselI[1, 2 x], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Formula

E.g.f. of column k: exp(k*x)*BesselI(1,2*x).

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

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)).
Showing 1-4 of 4 results.

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