A207612 Triangle of coefficients of polynomials u(n,x) jointly generated with A207613; see the Formula section.

1, 2, 4, 2, 7, 6, 4, 12, 14, 12, 8, 20, 30, 32, 24, 16, 33, 60, 76, 72, 48, 32, 54, 116, 168, 184, 160, 96, 64, 88, 218, 356, 440, 432, 352, 192, 128, 143, 402, 728, 1000, 1104, 992, 768, 384, 256, 232, 730, 1452, 2184, 2656, 2688, 2240, 1664, 768, 512

Offset: 1

Author

Clark Kimberling, Feb 19 2012

Keywords

• nonn
• tabf

Comments

Column 1: A000071

Column 2: 2*A023610

Examples

			First five rows:
1
2
4....2
7....6....4
12...14...12...8
		

Crossrefs

Cf. A207613.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x] + 1
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A207612 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A207613 *)

Python

from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + 2*x*v(n - 1, x) + 1
def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 28 2017

Formula

u(n,x)=u(n-1,x)+v(n-1,x), v(n,x)=u(n-1,x)+2x*v(n-1,x)+1,

where u(1,x)=1, v(1,x)=1.