A143506 -id:A143506 - cp's OEIS frontend

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.

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

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

Row sums yield A000670 (without leading 1).

			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

Eric Weisstein's World of Mathematics, Polylogarithm

Compare with A141720.
Cf. A008292.
Cf. A143506, A143507.

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

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

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

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

Row sums yield A144141.

			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

Wikipedia, Hermite polynomials

Cf. A060821.

Cf. A143505, A143506.

Table[CoefficientList[FullSimplify[x^n*HermiteH[n, x + 1/x]], x], {n,
  0, 10}]//Flatten

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

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

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

cp's OEIS Frontend

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.

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