A208510 Triangle of coefficients of polynomials u(n,x) jointly generated with A029653; see the Formula section.
1, 1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 7, 9, 5, 1, 1, 9, 16, 14, 6, 1, 1, 11, 25, 30, 20, 7, 1, 1, 13, 36, 55, 50, 27, 8, 1, 1, 15, 49, 91, 105, 77, 35, 9, 1, 1, 17, 64, 140, 196, 182, 112, 44, 10, 1, 1, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1, 1, 21, 100, 285, 540, 714, 672, 450, 210, 65, 12, 1
Offset: 1
Examples
First five rows: 1 1...1 1...3...1 1...5...4...1 1...7...9...5...1 First five polynomials u(n,x): 1 1 + x 1 + 3x + x^2 1 + 5x + 4x^2 + x^3 1 + 7x + 9x^2 + 5x^3 + x^4
Programs
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Mathematica
u[1, x_] := 1; v[1, x_] := 1; z = 16; u[n_, x_] := u[n - 1, x] + x*v[n - 1, x]; v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1; Table[Expand[u[n, x]], {n, 1, z/2}] Table[Expand[v[n, x]], {n, 1, z/2}] cu = Table[CoefficientList[u[n, x], x], {n, 1, z}]; TableForm[cu] Flatten[%] (* A208510 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A029653 *) -
Python
from sympy import Poly from sympy.abc import x def u(n, x): return 1 if n==1 else u(n - 1, x) + 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 27 2017
Formula
u(n,x)=u(n-1,x)+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.
Also, u(n,x)=(x+1)*u(n-1,x)+x for n>2, with u(n,2)=x+1.
Extensions
Corrected by Philippe Deléham, Apr 10 2012
Corrections and additions by Clark Kimberling, May 09 2012
Corrections in the overview by Georg Fischer, Sep 04 2021
Comments