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A193590 Augmentation of the Euler triangle A008292. See Comments.

%I A193590 #6 Mar 30 2012 18:57:38
%S A193590 1,1,1,1,5,2,1,16,33,8,1,42,275,342,58,1,99,1669,6441,5600,718,1,219,
%T A193590 8503,82149,217694,143126,14528,1,466,39076,843268,5466197,10792622,
%U A193590 5628738,466220,1,968,168786,7621160,107506633,509354984,788338180
%N A193590 Augmentation of the Euler triangle A008292.  See Comments.
%C A193590 For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
%C A193590 Regarding A193590, (column 1)=A002662, with general term 2^n-1-n(n+1)/2.
%e A193590 First 5 rows of A193589:
%e A193590 1
%e A193590 1....1
%e A193590 1....5....2
%e A193590 1....16...33....8
%e A193590 1....42...275...342....58
%t A193590 p[n_, k_] :=
%t A193590 Sum[((-1)^j)*((k + 1 - j)^(n + 1))*Binomial[n + 2, j], {j, 0, k + 1}]
%t A193590 (* A008292, Euler triangle *)
%t A193590 Table[p[n, k], {n, 0, 5}, {k, 0, n}]
%t A193590 m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
%t A193590 TableForm[m[4]]
%t A193590 w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
%t A193590 v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
%t A193590 v[n_] := v[n - 1].m[n]
%t A193590 TableForm[Table[v[n], {n, 0, 6}]]  (* A193590  *)
%t A193590 Flatten[Table[v[n], {n, 0, 8}]]
%Y A193590 Cf. A008292, A193091.
%K A193590 nonn,tabl
%O A193590 0,5
%A A193590 _Clark Kimberling_, Jul 31 2011