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A193860 Triangular array: the fission of ((2x+1)^n) by (q(n,x)), where q(n,x)=x^n+x^(n-1)+...+x+1.

%I A193860 #9 Jul 16 2013 10:00:27
%S A193860 1,1,5,1,7,19,1,9,33,65,1,11,51,131,211,1,13,73,233,473,665,1,15,99,
%T A193860 379,939,1611,2059,1,17,129,577,1697,3489,5281,6305,1,19,163,835,2851,
%U A193860 6883,12259,16867,19171,1,21,201,1161,4521,12585,26025,41385,52905
%N A193860 Triangular array:  the fission of ((2x+1)^n) by (q(n,x)), where q(n,x)=x^n+x^(n-1)+...+x+1.
%C A193860 See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.
%F A193860 From _Peter Bala_, Jul 16 2013: (Start)
%F A193860 T(n,k) = sum {i = 0..k} binomial(n+1,k-i)*2^(k-i) for 0 <= k <= n.
%F A193860 O.g.f.: 1/( (1 - 3*x*t)*(1 - (2*x + 1)*t) ) = 1 + (1 + 5*x)*t + (1 + 7*x + 19*x^2)*t^2 + ....
%F A193860 The n-th row polynomial R(n,x) = 1/(1 - x)*( (2*x + 1)^(n+1) - (3*x)^(n+1) ). Cf. A193823. (End)
%e A193860 First six rows:
%e A193860 1
%e A193860 1...5
%e A193860 1...7....19
%e A193860 1...9....33...65
%e A193860 1...11...51...131...211
%e A193860 1...13...73...233...473...665
%t A193860 z = 10;
%t A193860 p[n_, x_] := (2 x + 1)^n;
%t A193860 q[0, x_] := 1; q[n_, x_] := x*q[n - 1, x] + 1;
%t A193860 p1[n_, k_] := Coefficient[p[n, x], x^k];
%t A193860 p1[n_, 0] := p[n, x] /. x -> 0;
%t A193860 d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
%t A193860 h[n_] := CoefficientList[d[n, x], {x}]
%t A193860 TableForm[Table[Reverse[h[n]], {n, 0, z}]]
%t A193860 Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193860 *)
%t A193860 TableForm[Table[h[n], {n, 0, z}]]
%t A193860 Flatten[Table[h[n], {n, -1, z}]]   (* A193861  *)
%Y A193860 Cf. A193842, A193861. A193823.
%K A193860 nonn,tabl
%O A193860 0,3
%A A193860 _Clark Kimberling_, Aug 07 2011