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

1,3

Comments

Subtriangle of the triangle given by (1, 0, 2/3, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012

Links

Table of n, a(n) for n=1..66.

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

Example

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

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 *)

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 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

Crossrefs

Cf. A207608.

Sequence in context: A049541 A352939 A249757 * A322428 A130300 A366873

Adjacent sequences: A207606 A207607 A207608 * A207610 A207611 A207612

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 19 2012

Status

approved