A207537 Triangle of coefficients of polynomials u(n,x) jointly generated with A207538; see Formula section.

1, 2, 1, 4, 3, 8, 8, 1, 16, 20, 5, 32, 48, 18, 1, 64, 112, 56, 7, 128, 256, 160, 32, 1, 256, 576, 432, 120, 9, 512, 1280, 1120, 400, 50, 1, 1024, 2816, 2816, 1232, 220, 11, 2048, 6144, 6912, 3584, 840, 72, 1, 4096, 13312, 16640, 9984, 2912, 364, 13

Offset: 1

Author

Clark Kimberling, Feb 18 2012

Keywords

• nonn
• tabf

Comments

Another version in A201701. - Philippe Deléham, Mar 03 2012

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

Columns: A011782, A001792, A001793, A001794, A006974, A006975, A006976. - Philippe Deléham, Mar 03 2012

Diagonal sums: A052980. - Philippe Deléham, Mar 03 2012

Examples

			First seven rows:
   1;
   2,   1;
   4,   3;
   8,   8,  1;
  16,  20,  5,
  32,  48, 18, 1;
  64, 112, 56, 7;
From _Philippe Deléham_, Mar 03 2012: (Start)
Triangle A201701 begins:
   1;
   1,   0;
   2,   1,  0;
   4,   3,  0, 0;
   8,   8,  1, 0, 0;
  16,  20,  5, 0, 0, 0;
  32,  48, 18, 1, 0, 0, 0;
  64, 112, 56, 7, 0, 0, 0, 0;
  ... (End)
		

Crossrefs

Cf. A028297, A207538, A133156.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + 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[%]  (* A207537, |A028297| *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207538, |A133156| *)
(* Prepending 1 and with offset 0: *)
Tpoly[n_] := HypergeometricPFQ[{-n/2, -n/2 + 1/2}, {1/2}, x + 1];
Table[CoefficientList[Tpoly[n], x], {n, 0, 12}] // Flatten (* Peter Luschny, Feb 03 2021 *)

Formula

u(n,x) = u(n-1,x) + (x+1)*v(n-1,x), v(n,x) = u(n-1,x) + v(n-1,x), where u(1,x)=1, v(1,x)=1. Also, A207537 = |A028297|.

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

G.f.: -(1+x*y)*x*y/(-1+2*x+x^2*y). - R. J. Mathar, Aug 11 2015

T(n, k) = [x^k] hypergeom([-n/2, -n/2 + 1/2], [1/2], x + 1) provided offset is set to 0 and 1 prepended. - Peter Luschny, Feb 03 2021