A193590 - OEIS

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

%S 1,1,1,1,5,2,1,16,33,8,1,42,275,342,58,1,99,1669,6441,5600,718,1,219,

%T 8503,82149,217694,143126,14528,1,466,39076,843268,5466197,10792622,

%U 5628738,466220,1,968,168786,7621160,107506633,509354984,788338180

%N Augmentation of the Euler triangle A008292.  See Comments.

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

%C Regarding A193590, (column 1)=A002662, with general term 2^n-1-n(n+1)/2.

%e First 5 rows of A193589:

%e 1

%e 1....1

%e 1....5....2

%e 1....16...33....8

%e 1....42...275...342....58

%t p[n_, k_] :=

%t Sum[((-1)^j)*((k + 1 - j)^(n + 1))*Binomial[n + 2, j], {j, 0, k + 1}]

%t (* A008292, Euler 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, 6}]]  (* A193590  *)

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

%Y Cf. A008292, A193091.

%K nonn,tabl

%O 0,5

%A _Clark Kimberling_, Jul 31 2011