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

1, 1, 5, 1, 7, 19, 1, 9, 33, 65, 1, 11, 51, 131, 211, 1, 13, 73, 233, 473, 665, 1, 15, 99, 379, 939, 1611, 2059, 1, 17, 129, 577, 1697, 3489, 5281, 6305, 1, 19, 163, 835, 2851, 6883, 12259, 16867, 19171, 1, 21, 201, 1161, 4521, 12585, 26025, 41385, 52905

Offset: 0

Author

Clark Kimberling, Aug 07 2011

Keywords

• nonn
• tabl

Comments

See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.

Examples

			First six rows:
1
1...5
1...7....19
1...9....33...65
1...11...51...131...211
1...13...73...233...473...665
		

Crossrefs

Cf. A193842, A193861. A193823.

Programs

Mathematica

z = 10;
p[n_, x_] := (2 x + 1)^n;
q[0, x_] := 1; q[n_, x_] := x*q[n - 1, x] + 1;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193860 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]   (* A193861  *)

Formula

From Peter Bala, Jul 16 2013: (Start)

T(n,k) = sum {i = 0..k} binomial(n+1,k-i)*2^(k-i) for 0 <= k <= n.

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 + ....

The n-th row polynomial R(n,x) = 1/(1 - x)*( (2*x + 1)^(n+1) - (3*x)^(n+1) ). Cf. A193823. (End)