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%I #5 Mar 30 2012 18:57:38
%S 1,1,1,1,4,2,1,9,20,7,1,16,80,131,37,1,25,220,806,1085,265,1,36,490,
%T 3130,9360,10952,2402,1,49,952,9325,48224,124498,130852,26371,1,64,
%U 1680,23317,183569,813886,1876101,1809430,340272
%N Augmentation of the triangle A011973. See Comments.
%C For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
%C (col 2): A000290 (the squares)
%C (col 3)= 2*A002419
%e First five rows of A193607:
%e 1
%e 1...1
%e 1...4....2
%e 1...9....20...7
%e 1...16...80...131...37
%t p[n_, k_] := Binomial[2 n - k, k];
%t Table[p[n, k], {n, 0, 7}, {k, 0, n}] (* A011973 *)
%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, 9}]] (* A193607 *)
%t Flatten[Table[v[n], {n, 0, 8}]]
%Y Cf. A011973, A193091, A002419.
%K nonn,tabl
%O 0,5
%A _Clark Kimberling_, Jul 31 2011
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