cp's OEIS Frontend

A193093 Augmentation of the triangular array P=A094727 given by p(n,k)=n+k+1 for 0<=k<=n. See Comments.

%I A193093 #7 Jan 03 2014 15:58:36
%S A193093 1,2,3,6,14,19,24,72,130,169,120,432,918,1482,1877,720,3000,7224,
%T A193093 13140,19846,24675,5040,23760,63600,127104,210726,304006,372611,40320,
%U A193093 211680,622080,1350000,2412408,3754656,5234114,6340961,362880,2096640
%N A193093 Augmentation of the triangular array P=A094727 given by p(n,k)=n+k+1 for 0<=k<=n.  See Comments.
%C A193093 For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
%C A193093 Regarding W=A193093, we have w(n,0)=(n+1)! .
%e A193093 First 5 rows:
%e A193093 1
%e A193093 2.....3
%e A193093 6.....14....19
%e A193093 24....72....130....169
%e A193093 120...432....918...1482...1877
%t A193093 p[n_, k_] := n + k + 1
%t A193093 Table[p[n, k], {n, 0, 5}, {k, 0, n}] (* A094727 *)
%t A193093 m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
%t A193093 TableForm[m[4]]
%t A193093 w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
%t A193093 v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
%t A193093 v[n_] := v[n - 1].m[n]
%t A193093 TableForm[Table[v[n], {n, 0, 6}]] (* A193093 *)
%t A193093 Flatten[Table[v[n], {n, 0, 8}]]
%Y A193093 Cf. A094727.
%K A193093 nonn,tabl
%O A193093 0,2
%A A193093 _Clark Kimberling_, Jul 30 2011