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%I #6 Aug 19 2018 11:55:22
%S 1,1,0,1,1,1,1,4,4,0,1,11,17,7,1,1,26,76,66,16,0,1,57,317,467,237,31,
%T 1,1,120,1212,2962,2612,806,64,0,1,247,4321,17215,24145,13519,2641,
%U 127,1,1,502,14644,92554,199192,178486,65884,8434,256,0
%N Coefficients of the polynomials generated by the e.g.f. cosh(x*z)*(x-1)/(x-exp(z*(x-1))), triangle read by rows, T(n,k) for 0 <= k <= n.
%e [n\k][0, 1, 2, 3, 4, 5, 6, 7, 8]
%e [0] 1;
%e [1] 1, 0;
%e [2] 1, 1, 1;
%e [3] 1, 4, 4, 0;
%e [4] 1, 11, 17, 7, 1;
%e [5] 1, 26, 76, 66, 16, 0;
%e [6] 1, 57, 317, 467, 237, 31, 1;
%e [7] 1, 120, 1212, 2962, 2612, 806, 64, 0;
%e [8] 1, 247, 4321, 17215, 24145, 13519, 2641, 127, 1;
%p gf := cosh(x*z)*(x-1)/(x-exp(z*(x-1))):
%p ser := series(gf, z, 12): p := n -> normal(n!*coeff(ser, z, n)):
%p seq(seq(coeff(p(n),x,k), k=0..n), n=0..10);
%Y Row sums are (-1)^n*A009179(n).
%Y Alternating row sums are 1.
%Y Polynomials evaluated at x = 0 are 1.
%Y T(n, n-1) = A051049(n-1) for n >= 1.
%Y T(n, 1) = A000295(n) for n >= 0.
%Y Cf. A046802, A173018.
%K nonn,tabl
%O 0,8
%A _Peter Luschny_, Aug 19 2018
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