A207608 Triangle of coefficients of polynomials u(n,x) jointly generated with A207609; see the Formula section.

1, 2, 3, 3, 4, 11, 3, 5, 26, 20, 3, 6, 50, 74, 29, 3, 7, 85, 204, 149, 38, 3, 8, 133, 469, 547, 251, 47, 3, 9, 196, 952, 1618, 1160, 380, 56, 3, 10, 276, 1764, 4110, 4234, 2124, 536, 65, 3, 11, 375, 3048, 9318, 13036, 9262, 3520, 719, 74, 3, 12, 495, 4983

Offset: 1

Author

Clark Kimberling, Feb 19 2012

Keywords

• nonn
• tabf

Comments

As triangle T(n,k) with 0<=k<=n and with zeros omitted, it is (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012

Examples

			First five rows:
  1;
  2;
  3,  3;
  4, 11,  3;
  5, 26, 20,  3;
Triangle (2, -1/2, 1/2, 0, 0, 0, 0, ...) DELTA (0, 3/2, -1/2, 0, 0, 0, 0, ...) begins:
  1;
  2,   0;
  3,   3,   0;
  4,  11,   3,   0;
  5,  26,  20,   3,   0;
  6,  50,  74,  29,   3,   0;
  7,  85, 204, 149,  38,   3,   0;
  ... - _Philippe Deléham_, Mar 03 2012
		

Crossrefs

Cf. A207609.

Programs

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(u(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.

From Philippe Deléham, Mar 03 2012: (Start)

As triangle T(n,k), 0 <= k <= n:

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-1,k) + T(n-2,k-1) with T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n,k) = 0 if k < 0 or if k > n.

G.f.: (1-y*x)/(1 - (2+y)*x - (y-1)*x^2).

Sum_{k=0..n} T(n,k)*x^k = A000027(n+1), A025192(n), A001077(n), A180038(n) for x = 0, 1, 2, 3 respectively. (End)