A142175 - OEIS

A142175

T(n,k) = (1/4)*A007318(n,k) - (3/2)*A008292(n+1,k+1) + (9/4)*A060187(n+1,k+1), triangle read by rows (0 <= k <= n).

1, 1, 1, 1, 8, 1, 1, 36, 36, 1, 1, 133, 420, 133, 1, 1, 449, 3334, 3334, 449, 1, 1, 1446, 21939, 49364, 21939, 1446, 1, 1, 4534, 130044, 560957, 560957, 130044, 4534, 1, 1, 13991, 724222, 5459561, 10284514, 5459561, 724222, 13991, 1, 1, 42747, 3880014

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OFFSET

0,5

COMMENTS

Row n gives the coefficients in the expansion of (1/4)*(1 + x)^n + (9/4)*2^n*(1 - x)^(1 + n)*Phi(x, -n, 1/2) - (3/2)*(1 - x)^(n + 2)*Phi(x, -1 - n, 1), where Phi is the Lerch transcendant.

LINKS

Table of n, a(n) for n=0..47.

Wikipedia, Lerch zeta function

Wikipedia, Polylogarithm

FORMULA

E.g.f.: (exp((1 + x)*y) - 6*(1 - x)^2*exp(y*(1 - x))/(1 - x*exp(y*(1 - x)))^2 + 9*(1 - x)*exp((1 - x)*y)/(1 - x*exp(2*(1 - x)*y)))/4. - Franck Maminirina Ramaharo, Oct 20 2018

EXAMPLE

Triangle begins:

1, 1;

1, 8, 1;

1, 36, 36, 1;

1, 133, 420, 133, 1;

1, 449, 3334, 3334, 449, 1;

1, 1446, 21939, 49364, 21939, 1446, 1;

1, 4534, 130044, 560957, 560957, 130044, 4534, 1;

... reformatted. - Franck Maminirina Ramaharo, Oct 21 2018

MATHEMATICA

p[x_, n_] = 1/4*(1 + x)^n + 9/4*2^n*(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2] - 3/2*(1 - x)^(2 + n)*PolyLog[-1 - n, x]/x;