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

1,2

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

Links

Table of n, a(n) for n=1..59.

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)

Example

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

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(u(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 28 2017

Crossrefs

Cf. A207609.

Sequence in context: A173590 A128744 A293984 * A290818 A240376 A118963

Adjacent sequences: A207605 A207606 A207607 * A207609 A207610 A207611

Keyword

nonn,tabf

Author

Clark Kimberling, Feb 19 2012

Status

approved