A193603 - OEIS

%I #6 Mar 30 2012 18:57:38

%S 1,1,2,1,5,8,1,9,30,44,1,14,77,212,296,1,20,163,700,1712,2312,1,27,

%T 305,1877,6882,15476,20384,1,35,523,4365,22380,73240,154424,199376,1,

%U 44,840,9134,62479,280630,841312,1683992,2138336

%N Augmentation of the triangle A008949.  See Comments.

%C For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.  The right edge of the triangle A193603 is A111537.

%e First five rows of A193603:

%e 1

%e 1...2

%e 1...5....8

%e 1...9...30....44

%e 1...14...77..212...296

%t p[n_, k_] := Sum[Binomial[n, h], {h, 0, k}] (* A008949 *)

%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, 6}]] (* A193603 *)

%t Flatten[Table[v[n], {n, 0, 8}]]

%Y Cf. A008949, A193091, A111537.

%K nonn,tabl

%O 0,3

%A _Clark Kimberling_, Jul 31 2011