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%I #14 Nov 14 2012 15:28:38
%S 1,1,1,-1,1,5,-2,-2,1,1,15,13,-19,3,3,-1,1,37,128,-26,-74,46,-4,-4,1,
%T 1,83,679,755,-654,-68,230,-90,5,5,-1,1,177,2866,9048,2091,-5741,1856,
%U 498,-545,155,-6,-6,1,1,367,10721,67541,98069,-24675,-35027,22717,-3773,-1673,1099,-245,7,7,-1,1
%N Triangle, read by rows, where T(n,k) is defined for n>=1, k=1..2*n-1, by a formula analogous to the second-order Eulerian triangle A008517.
%C Compare to the o.g.f. of row n, E2(x,n), in the second-order Eulerian triangle A008517:
%C E2(x,n) = (1-x)^(2*n+1) * Sum_{k>=0} k^n * k^k * exp(-k*x) * x^k/k!.
%H Paul D. Hanna, <a href="/A219120/b219120.txt">Rows n = 1..32, flattened.</a>
%F O.g.f. of row n, R(x,n), is given by:
%F R(x,n) = (1-x)^(2*n-1) * Sum_{k>=0} k^n *(k+1)^(k-1) * exp(-(k+1)*x) * x^k/k!.
%F Row sums = A001147, which is the odd double factorials.
%F Column 1 = A050488(n-1), where A050488(n) = 3*(2^n-1) - 2*n.
%F Central terms of rows = A219121.
%e Triangle begins:
%e 1;
%e 1, 1, -1;
%e 1, 5, -2, -2, 1;
%e 1, 15, 13, -19, 3, 3, -1;
%e 1, 37, 128, -26, -74, 46, -4, -4, 1;
%e 1, 83, 679, 755, -654, -68, 230, -90, 5, 5, -1;
%e 1, 177, 2866, 9048, 2091, -5741, 1856, 498, -545, 155, -6, -6, 1;
%e 1, 367, 10721, 67541, 98069, -24675, -35027, 22717, -3773, -1673, 1099, -245, 7, 7, -1;
%e 1, 749, 37300, 409170, 1290116, 863168, -590008, -108832, 182806, -65858, 5824, 4228, -1988, 364, -8, -8, 1; ...
%o (PARI) {T(n,k)=polcoeff((1-x)^(2*n-1)*sum(j=0,2*n,(j^n)*(j+1)^(j-1)*x^j/j!*exp(-(j+1)*x +O(x^k))),k)}
%o for(n=1,10,for(k=1,2*n-1,print1(T(n,k),", "));print(""))
%Y Cf. A219121, A217900, A008517, A001147.
%K sign,tabf
%O 1,6
%A _Paul D. Hanna_, Nov 12 2012
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