A193630 - OEIS

%I #7 Mar 30 2012 18:57:38

%S 1,1,2,1,5,7,1,9,28,33,1,14,74,181,191,1,20,159,637,1333,1297,1,27,

%T 300,1767,5906,11029,10063,1,35,517,4190,20256,59324,101351,87669,1,

%U 44,833,8873,58339,244125,645146,1024949,847015,1,54,1274,17241,147680

%N Augmentation of the triangle A074909.  See Comments.

%C For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

%C Regarding A193630, writing the general term as w(n,k),

%C w(n,n):  A104981

%C w(n,n-1):  A156629

%e First five rows of A193607:

%e 1

%e 1...2

%e 1...5....7

%e 1...9....28...33

%e 1...14...74...181...191

%t p[n_, k_] := Binomial[n + 1, k];

%t Table[p[n, k], {n, 0, 7}, {k, 0, n}]  (* A074909 *)

%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}]] (* A193630 *)

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

%Y Cf. A193091, A074909.

%K nonn,tabl

%O 0,3

%A _Clark Kimberling_, Aug 01 2011