A207609 Triangle of coefficients of polynomials v(n,x) jointly generated with A207608; see Formula section.
1, 1, 3, 1, 8, 3, 1, 15, 17, 3, 1, 24, 54, 26, 3, 1, 35, 130, 120, 35, 3, 1, 48, 265, 398, 213, 44, 3, 1, 63, 483, 1071, 909, 333, 53, 3, 1, 80, 812, 2492, 3074, 1744, 480, 62, 3, 1, 99, 1284, 5208, 8802, 7138, 2984, 654, 71, 3, 1, 120, 1935, 10020, 22230, 24408, 14370, 4710, 855, 80, 3
Offset: 1
Examples
First five rows: 1 1...3 1...8....3 1...15...17...3 1...24...54...26...3 Triangle (1, 0, 2/3, 1/3, 0, 0, 0, ...) DELTA (0, 3, -2, 0, 0, 0, ...) begins : 1 1, 0 1, 3, 0 1, 8, 3, 0 1, 15, 17, 3, 0 1, 24, 54, 26, 3, 0 1, 35, 130, 120, 35, 3, 0
Crossrefs
Cf. A207608.
Programs
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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_] := 2 x*u[n - 1, x] + (x + 1) v[n - 1, x] 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[%] (* A207608 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A207609 *) -
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 2*x*u(n - 1, x) + (x + 1)*v(n - 1, x) def a(n): return Poly(v(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)=2x*u(n-1,x)+(x+1)v(n-1,x),
where u(1,x)=1, v(1,x)=1.
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) - T(n-2,k), n>2. - Philippe Deléham, Mar 03 2012
Sum_{k, 0<=k<=n, n>=1} T(n,k)*x^k = A000012(n), A052156(n-1), A048876(n-1) for x = 0, 1, 2 respectively. - Philippe Deléham, Mar 03 2012
G.f.: -(1-x+2*x*y)*x*y/(-1+2*x+x*y+x^2*y-x^2). - R. J. Mathar, Aug 11 2015
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