A143505 -id:A143505 - cp's OEIS frontend

A143506 Irregular triangle read by rows: first row is 1, and n-th row gives the coefficients of x^(n - 1)*R(n,x + 1/x)/(x + 1/x), where R(n,x) is the n-th row polynomial for A060187.

1, 1, 1, 1, 1, 6, 3, 6, 1, 1, 23, 26, 47, 26, 23, 1, 1, 76, 234, 304, 467, 304, 234, 76, 1, 1, 237, 1687, 2630, 5293, 4787, 5293, 2630, 1687, 237, 1, 1, 722, 10549, 27158, 52730, 78586, 84365, 78586, 52730, 27158, 10549, 722, 1, 1, 2179, 60664, 272797, 563029, 1132234

Offset: 0

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

Row sums yield A080253.

			Triangle begins:
   1;
   1,   1,    1;
   1,   6,    3,    6,    1;
   1,  23,   26,   47,   26,   23,    1;
   1,  76,  234,  304,  467,  304,  234,   76,    1;
   1, 237, 1687, 2630, 5293, 4787, 5293, 2630, 1687, 237, 1;
    ... reformatted. - _Franck Maminirina Ramaharo_, Oct 25 2018

Wikipedia, Lerch zeta function

Cf. A008292, A060187.

Cf. A143505, A143507.

Table[CoefficientList[FullSimplify[ExpandAll[2^n*(1 - x - 1/x)^(1 + n)*x^n*LerchPhi[x + 1/x, -n, 1/2]]], x], {n, 0, 10}]//Flatten

Row n is generated by the polynomial 2^n*(1 - x - 1/x)^(1 + n)*x^n*Phi(x + 1/x, -n, 1/2), where Phi is the Lerch transcendant.

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

Edited, new name, and offset corrected 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

A143506 Irregular triangle read by rows: first row is 1, and n-th row gives the coefficients of x^(n - 1)*R(n,x + 1/x)/(x + 1/x), where R(n,x) is the n-th row polynomial for A060187.

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