A160232 Array read by antidiagonals: row n has g.f. ((1-x)/(1-2x))^n.

1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 1, 4, 9, 12, 8, 1, 5, 14, 25, 28, 16, 1, 6, 20, 44, 66, 64, 32, 1, 7, 27, 70, 129, 168, 144, 64, 1, 8, 35, 104, 225, 360, 416, 320, 128, 1, 9, 44, 147, 363, 681, 968, 1008, 704, 256, 1, 10, 54, 200, 553, 1182, 1970, 2528, 2400, 1536, 512, 1, 11, 65

Offset: 1

Author

N. J. A. Sloane, May 15 2010

Keywords

• nonn
• tabl

Comments

Suggested by a question from Phyllis Chinn (Humboldt State University).

As triangle, mirror image of A105306. - Philippe Deléham, Nov 01 2011

A160232 is jointly generated with A208341 as a triangular array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n > 1, u(n,x) = u(n-1,x) + x*v(n-1)x and v(n,x) = u(n-1,x) + 2x*v(n-1,x). See the Mathematica section. - Clark Kimberling, Feb 25 2012

Subtriangle of the triangle T(n,k) given by (1, 0, 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 08 2012

Examples

			Array begins:
  1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, ...
  1, 2, 5, 12, 28, 64, 144, 320, 704, 1536, 3328, 7168, 15360, 32768, 69632, 147456, 311296, 655360, 1376256, ...
  1, 3, 9, 25, 66, 168, 416, 1008, 2400, 5632, 13056, 29952, 68096, 153600, 344064, 765952, 1695744, 3735552, ...
  1, 4, 14, 44, 129, 360, 968, 2528, 6448, 16128, 39680, 96256, 230656, 546816, 1284096, 2990080, 6909952, ...
  1, 5, 20, 70, 225, 681, 1970, 5500, 14920, 39520, 102592, 261760, 657920, 1632000, 4001280, 9708544, ...
  1, 6, 27, 104, 363, 1182, 3653, 10836, 31092, 86784, 236640, 632448, 1661056, 4296192, 10961664, 27630592, ...
From _Clark Kimberling_, Feb 25 2012: (Start)
As a triangle (see Comments):
  1;
  1,  1;
  1,  2,  2;
  1,  3,  5,  4;
  1,  4,  9, 12,  8;  (End)
From _Philippe Deléham_, Mar 08 2012: (Start)
(1, 0, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, ...) begins:
  1;
  1,  0;
  1,  1,  0;
  1,  2,  2,  0;
  1,  3,  5,  4,  0;
  1,  4,  9, 12,  8,  0;
  1,  5, 14, 25, 28, 16,  0; (End)
		

Crossrefs

Rows give A011782, A045623, A058396, A062109, A169792-A169797.

Cf. A062110, A105306, A208341.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 2*x*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[%]  (* A160232 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208341 *)
(* Clark Kimberling, Feb 25 2012 *)

Formula

From Philippe Deléham, Mar 08 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(0,0) = 1, T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k < 0 or if k > n.

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

Sum_{k=0..n, n>0} T(n,k)*x^k = A000012(n), A001519(n), A052984(n-1) for x = 0, 1, 2 respectively. (End)