A207610 Triangle of coefficients of polynomials u(n,x) jointly generated with A207611; see the Formula section.

1, 2, 4, 1, 7, 3, 1, 12, 7, 3, 1, 20, 15, 8, 3, 1, 33, 30, 19, 9, 3, 1, 54, 58, 42, 23, 10, 3, 1, 88, 109, 89, 55, 27, 11, 3, 1, 143, 201, 182, 125, 69, 31, 12, 3, 1, 232, 365, 363, 273, 166, 84, 35, 13, 3, 1, 376, 655, 709, 579, 383, 212, 100, 39, 14, 3, 1, 609

Offset

1,2

Links

Table of n, a(n) for n=1..68.

Formula

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

Example

First five rows:
1
2
4...1
7...3...1
12...7...3...1

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] + 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[%]    (* A207610 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A207611 *)

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) + 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. A207611.

Cf. A000071 (column 1), A023610 (column 2).

Sequence in context: A302541 A119303 A256107 * A207616 A105552 A112852

Adjacent sequences: A207607 A207608 A207609 * A207611 A207612 A207613

Keyword

nonn,tabf

Author

Clark Kimberling, Feb 19 2012

Status

approved