A207606 Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.

1, 2, 3, 2, 4, 7, 2, 5, 16, 11, 2, 6, 30, 36, 15, 2, 7, 50, 91, 64, 19, 2, 8, 77, 196, 204, 100, 23, 2, 9, 112, 378, 540, 385, 144, 27, 2, 10, 156, 672, 1254, 1210, 650, 196, 31, 2, 11, 210, 1122, 2640, 3289, 2366, 1015, 256, 35, 2, 12, 275, 1782, 5148, 8008

Offset: 1

Author

Clark Kimberling, Feb 19 2012

Keywords

• nonn
• tabf

Comments

As triangle T(n,k) with 0 <= k <= n, it is (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 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,  2;
  4,  7,  2;
  5, 16, 11,  2;
Triangle (2, -1/2, 1/2, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, ...), 0 <= k <= n, begins:
  1;
  2,   0;
  3,   2,   0;
  4,   7,   2,   0;
  5,  16,  11,   2,   0;
  6,  30,  36,  15,   2,   0;
  7,  50,  91,  64,  19,   2,   0;
  8,  77, 196, 204, 100,  23,   2,   0;
		

Links

• G. C. Greubel, Rows n = 1..101 of the triangle, flattened

Crossrefs

Cf. A207607.

Programs

Maple

T:= proc(n, k) option remember;
      if k<0 or k>n then 0
    elif k=0 then n+2
    elif k=n then 2
    else 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k)
      fi; end:
1, seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Mar 15 2020

Mathematica

(* First program *)
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := 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[%]  (* A207606 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207607 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, n+2, If[k==n, 2, 2*T[n-1, k] - T[n-2, k] + T[n-1, k-1] ]]]; Join[{1}, Table[T[n, k], {n, 0, 10}, {k, 0, n}]]//Flatten (* G. C. Greubel, Mar 15 2020 *)

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 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

Sage

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==1): return n+1
    elif (k==n): return 2
    else: return 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k)
[1]+[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Mar 15 2020

Formula

u(n,x) = u(n-1,x) + v(n-1,x), v(n,x) = x*u(n-1,x) + (x+1)v(n-1,x), where u(1,x)=1, v(1,x)=1.

As triangle T(n,k) with 0 <= k <= n: g.f.: (1-y*x)/(1-(2+y)*x+x^2). - Philippe Deléham, Mar 03 2012

As triangle T(n,k) with 0 <= k <= n: Sum_{k=0..n} T(n,k)*x^k = A132677(n), A000034(n)*A057077(n), A057079(n), A000027(n+1), A001519(n+1), A001075(n), A002310(n), A038725(n), A172968(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5 respectively. - Philippe Deléham, Mar 03 2012

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k). - Philippe Deléham, Mar 03 2012

T(n,k) = C(n+k-1,2*k+1) + 2*C(n+k-1,2*k), where C is binomial. - Yuchun Ji, May 23 2019

T(n,k) = T(n-1,k) + A207607(n-1,k). - Yuchun Ji, May 28 2019