A112351 Triangle read by rows, generated from (..., 5, 3, 1).

1, 1, 3, 1, 6, 5, 1, 9, 19, 7, 1, 12, 42, 44, 9, 1, 15, 74, 138, 85, 11, 1, 18, 115, 316, 363, 146, 13, 1, 21, 165, 605, 1059, 819, 231, 15, 1, 24, 224, 1032, 2470, 2984, 1652, 344, 17, 1, 27, 292, 1624, 4974, 8378, 7380, 3060

Offset: 0

Author

Gary W. Adamson, Sep 05 2005

Keywords

• nonn
• tabl

Comments

A039755 (Analogs of a Stirling number of the second kind triangle) is generated through an analogous set of operations (but using the matrix M = [1 / 1 3 / 1 3 5 /...]). First few rows of the array are 1, 3, 5, 7, 9, 11, ...; 1, 6, 19, 44, 85, ...; 1, 9, 42, 138, 363, ...; 1, 12, 74, 316, 1059, ....

A112351 is jointly generated with A209414 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x) = x*u(n-1,x) + v(n-1,x) and v(n,x) = 2x*u(n-1,x) + (x+1)*v(n-1,x). See the Mathematica and Example sections. - Clark Kimberling, Mar 09 2012

Subtriangle of the triangle T(n,k) given by (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -4/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 12 2012

Examples

			The antidiagonal 1 9 19 7 of the array becomes row 3 of the triangle.
From _Clark Kimberling_, Mar 09 2012: (Start)
When jointly generated with A209414, the format as a triangle has the following first five rows:
  1;
  1,  3;
  1,  6,  5;
  1,  9, 19,   7;
  1, 12, 42,  44,  9;
  1, 15, 74, 138, 85, 11;
The corresponding first five polynomials are
  1,
  1 + 3x,
  1 + 6x + 5x^2,
  1 + 9x + 19x^2 + 7x^3,
  1 + 12x + 42x^2 + 44x^3 + 9x^4. (End)
(1, 0, 0, 0, 0, ...) DELTA (0, 3, -4/3, 1/3, 0, 0, 0, ...) begins:
  1;
  1,  0;
  1,  3,   0;
  1,  6,   5,   0;
  1,  9,  19,   7,   0;
  1, 12,  42,  44,   9,   0;
  1, 15,  74, 138,  85,  11,  0;
  1, 18, 115, 316, 363, 146, 13, 0;
- _Philippe Deléham_, Mar 12 2012
		

Crossrefs

Cf. A039755, A005900 (array row 2), A061927 (array row 3), A209414.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := 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[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209414 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A112351 *)
(* Clark Kimberling, Mar 09 2012 *)

Formula

Let M = an infinite lower triangular matrix of the form [1 / 3 1 / 5 3 1 / ...] (with the rest of the terms zeros). Perform M^n * [1 0 0 0 ...] forming an array. Antidiagonals of the array become rows of the triangle A112351.

From Philippe Deléham, Mar 12 2012: (Start)

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

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,1) = 0, T(2,1) = 3 and T(n,k) = 0 if k < 0 or if k > n.

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