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.

A143505 Triangle of coefficients of the polynomials x^(n - 1)*A(n,x + 1/x), where A(n,x) are the Eulerian polynomials of A008292.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 3, 4, 1, 1, 11, 14, 23, 14, 11, 1, 1, 26, 70, 104, 139, 104, 70, 26, 1, 1, 57, 307, 530, 973, 947, 973, 530, 307, 57, 1, 1, 120, 1197, 3016, 5970, 8568, 9549, 8568, 5970, 3016, 1197, 120, 1, 1, 247, 4300, 17101, 37105, 70474, 90069, 107241, 90069
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 25 2008

Keywords

Comments

Row sums yield A000670 (without leading 1).

Examples

			Triangle begins:
   1;
   1,  1,   1;
   1,  4,   3,   4,   1;
   1, 11,  14,  23,  14,  11,   1;
   1, 26,  70, 104, 139, 104,  70,  26,   1;
   1, 57, 307, 530, 973, 947, 973, 530, 307, 57, 1;
    ... reformatted. - _Franck Maminirina Ramaharo_, Oct 25 2018
		

Crossrefs

Compare with A141720.
Cf. A008292.

Programs

  • Mathematica
    Table[CoefficientList[FullSimplify[ExpandAll[(1 - x - 1/x)^(n + 1)*x^(n - 1)*PolyLog[-n, x + 1/x]/(x + 1/x)]], x], {n, 1, 10}]//Flatten

Formula

Row n is generated by the polynomial (1 - x - 1/x)^(n + 1)*x^(n - 1)*Li(-n, x + 1/x)/(x + 1/x), where Li(n, z) is the polylogarithm function.
E.g.f.: (exp(x*y) - exp((1 + x^2)*y))/(x*exp((1 + x^2)*y) - (1 + x^2)*exp(x*y)). - Franck Maminirina Ramaharo, Oct 25 2018

Extensions

Edited and new name by Franck Maminirina Ramaharo, Oct 25 2018

A143507 Triangle of coefficients of x^n*H_n(x + 1/x), where H_n(x) is the Hermite polynomial of order n.

Original entry on oeis.org

1, 2, 0, 2, 4, 0, 6, 0, 4, 8, 0, 12, 0, 12, 0, 8, 16, 0, 16, 0, 12, 0, 16, 0, 16, 32, 0, 0, 0, -40, 0, -40, 0, 0, 0, 32, 64, 0, -96, 0, -240, 0, -280, 0, -240, 0, -96, 0, 64, 128, 0, -448, 0, -672, 0, -560, 0, -560, 0, -672, 0, -448, 0, 128, 256, 0, -1536, 0, -896, 0, 896, 0, 1680, 0, 896, 0, -896, 0, -1536, 0, 256, 512, 0, -4608, 0, 512
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 25 2008

Keywords

Comments

Row sums yield A144141.

Examples

			Triangle begins:
     1;
     2, 0,    2;
     4, 0,    6, 0,    4;
     8, 0,   12, 0,   12, 0,    8;
    16, 0,   16, 0,   12, 0,   16, 0,   16;
    32, 0,    0, 0,  -40, 0,  -40, 0,    0, 0,   32;
    64, 0,  -96, 0, -240, 0, -280, 0, -240, 0,  -96, 0,   64;
   128, 0, -448, 0, -672, 0, -560, 0, -560, 0, -672, 0, -448, 0, 128;
    ... reformatted. - _Franck Maminirina Ramaharo_, Oct 25 2018
		

Crossrefs

Programs

  • Mathematica
    Table[CoefficientList[FullSimplify[x^n*HermiteH[n, x + 1/x]], x], {n,
      0, 10}]//Flatten
  • PARI
    row(n) = Vec(x^n*subst(polhermite(n,x),x,x+1/x));
    for (n=0, 10, print(row(n))); \\ Michel Marcus, Oct 27 2018

Formula

E.g.f.: exp(2*(1 + x^2)*y - x^2*y^2). - Franck Maminirina Ramaharo, Oct 25 2018

Extensions

Edited, new name and offset corrected by Franck Maminirina Ramaharo, Oct 25 2018
Showing 1-2 of 2 results.