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A193591 Augmentation of the Euler partition triangle A026820. See Comments.

%I A193591 #10 Oct 22 2024 05:38:39
%S A193591 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 A193591 652,2057,4005,5953,1,23,228,1395,5251,15004,27836,40668,1,28,355,
%U A193591 2555,13023,43470,115110,205516,295922,1,34,509,4689,27327,122006,371942
%N A193591 Augmentation of the Euler partition triangle A026820.  See Comments.
%C A193591 For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
%e A193591 First 5 rows:
%e A193591   1
%e A193591   1...2
%e A193591   1...4...7
%e A193591   1...7...19...31
%e A193591   1...10..45...103...161
%t A193591 p[n_, k_] := Length@IntegerPartitions[n + 1,
%t A193591    k + 1] (* A026820, Euler partition triangle *)
%t A193591 Table[p[n, k], {n, 0, 5}, {k, 0, n}]
%t A193591 m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
%t A193591 TableForm[m[4]]
%t A193591 w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
%t A193591 v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
%t A193591 v[n_] := v[n - 1].m[n]
%t A193591 TableForm[Table[v[n], {n, 0, 12}]]  (* A193591 *)
%t A193591 Flatten[Table[v[n], {n, 0, 9}]]
%Y A193591 Cf. A014616 (column 1), A026820, A193091.
%K A193591 nonn,tabl
%O A193591 0,3
%A A193591 _Clark Kimberling_, Jul 31 2011