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%I #5 Mar 30 2012 18:57:38
%S 1,1,2,1,4,7,1,7,19,31,1,10,45,103,161,1,14,82,297,617,937,1,18,146,
%T 652,2057,4005,5953,1,23,228,1395,5251,15004,27836,40668,1,28,355,
%U 2555,13023,43470,115110,205516,295922,1,34,509,4689,27327,122006,371942
%N Augmentation of the Euler partition triangle A026820. See Comments.
%C For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
%C Regarding A193591, column 1)=A014616.
%e First 5 rows of A193589:
%e 1
%e 1...2
%e 1...4...7
%e 1...7...19...31
%e 1...10..45...103...161
%t p[n_, k_] := Length@IntegerPartitions[n + 1,
%t k + 1] (* A026820, Euler partition triangle *)
%t Table[p[n, k], {n, 0, 5}, {k, 0, n}]
%t m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
%t TableForm[m[4]]
%t w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
%t v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
%t v[n_] := v[n - 1].m[n]
%t TableForm[Table[v[n], {n, 0, 12}]] (* A193591 *)
%t Flatten[Table[v[n], {n, 0, 9}]]
%Y Cf. A026820, A193091.
%K nonn,tabl
%O 0,3
%A _Clark Kimberling_, Jul 31 2011
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