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

1,2

Comments

Column 1: A000071

Column 2: 2*A023610

Links

Table of n, a(n) for n=1..56.

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.

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A207613.

Sequence in context: A110925 A214789 A207631 * A207620 A354766 A207622

Adjacent sequences: A207609 A207610 A207611 * A207613 A207614 A207615

Keyword

nonn,tabf

Author

Clark Kimberling, Feb 19 2012

Status

approved