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A326485 T(n, k) = 2^A050605(n) * n! * [x^k] [z^n] (4*exp(x*z))/(exp(z) + 1)^2, triangle read by rows, for 0 <= k <= n.

%I A326485 #10 Jul 13 2019 00:49:41
%S A326485 1,-1,1,1,-4,2,1,3,-6,2,-1,2,3,-4,1,-1,-5,5,5,-5,1,17,-24,-60,40,30,
%T A326485 -24,4,17,119,-84,-140,70,42,-28,4,-31,34,119,-56,-70,28,14,-8,1,-31,
%U A326485 -279,153,357,-126,-126,42,18,-9,1,691,-620,-2790,1020,1785,-504,-420,120,45,-20,2
%N A326485 T(n, k) = 2^A050605(n) * n! * [x^k] [z^n] (4*exp(x*z))/(exp(z) + 1)^2, triangle read by rows, for 0 <= k <= n.
%C A326485 These are the coefficients of the generalized Euler polynomials (case m=2) with a different normalization. See A326480 for further comments.
%e A326485 Triangle starts:
%e A326485 [0] [  1]
%e A326485 [1] [ -1,    1]
%e A326485 [2] [  1,   -4,   2]
%e A326485 [3] [  1,    3,  -6,    2]
%e A326485 [4] [ -1,    2,   3,   -4,    1]
%e A326485 [5] [ -1,   -5,   5,    5,   -5,    1]
%e A326485 [6] [ 17,  -24, -60,   40,   30,  -24,   4]
%e A326485 [7] [ 17,  119, -84, -140,   70,   42, -28,  4]
%e A326485 [8] [-31,   34, 119,  -56,  -70,   28,  14, -8,  1]
%e A326485 [9] [-31, -279, 153,  357, -126, -126,  42, 18, -9, 1]
%p A326485 E2n := proc(n) (4*exp(x*z))/(exp(z) + 1)^2;
%p A326485 series(%, z, 48); 2^A050605(n)*n!*coeff(%, z, n) end:
%p A326485 for n from 0 to 9 do PolynomialTools:-CoefficientList(E2n(n), x) od;
%Y A326485 Cf. A326480, A050605.
%K A326485 sign,tabl
%O A326485 0,5
%A A326485 _Peter Luschny_, Jul 12 2019