A094250 Array, A(n, k) = ((n+2)^(k+1) + (k+1)*n*(n+1) - 1)/(n+1)^2, read by antidiagonals.

1, 1, 3, 1, 3, 7, 1, 3, 8, 15, 1, 3, 9, 22, 31, 1, 3, 10, 31, 63, 63, 1, 3, 11, 42, 117, 185, 127, 1, 3, 12, 55, 199, 459, 550, 255, 1, 3, 13, 70, 315, 981, 1825, 1644, 511, 1, 3, 14, 87, 471, 1871, 4888, 7287, 4925, 1023, 1, 3, 15, 106, 673, 3273, 11203, 24420, 29133, 14767, 2047

Offset: 0

Author

N. J. A. Sloane, Jun 02 2004

Keywords

• nonn
• tabl

Examples

			Array, A(n, k), begins:
  1, 3,  7, 15,  31,   63,   127,    255,     511, ... A000225;
  1, 3,  8, 22,  63,  185,   550,   1644,    4925, ... A047926;
  1, 3,  9, 31, 117,  459,  1825,   7287,   29133, ... A073724;
  1, 3, 10, 42, 199,  981,  4888,  24420,  122077, ... A094195;
  1, 3, 11, 55, 315, 1871, 11203,  67191,  403115, ... A094259;
  1, 3, 12, 70, 471, 3273, 22882, 160140, 1120941, ...
Antidiagonals, T(n, k), begins as:
  1;
  1, 3;
  1, 3,  7;
  1, 3,  8, 15;
  1, 3,  9, 22,  31;
  1, 3, 10, 31,  63,   63;
  1, 3, 11, 42, 117,  185,  127;
  1, 3, 12, 55, 199,  459,  550,  255;
  1, 3, 13, 70, 315,  981, 1825, 1644,  511;
  1, 3, 14, 87, 471, 1871, 4888, 7287, 4925, 1023;
		

Links

• G. C. Greubel, Antidiagonals n = 0..50, flattened

Crossrefs

Rows are A000225, A047926, A073724, A094195, A094259.

Programs

Magma

A094250:= func< n,k | ((n-k+2)^(k+1) + (k+1)*(n-k)*(n-k+1) - 1)/(n-k+1)^2 >;
[A094250(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Aug 18 2023

Mathematica

A094250[n_, k_]:= ((n-k+2)^(k+1) + (k+1)*(n-k)*(n-k+1) - 1)/(n-k+1)^2;
Table[A094250[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Aug 18 2023 *)

SageMath

def A094250(n, k): return ((n-k+2)^(k+1) + (k+1)*(n-k)*(n-k+1) - 1)/(n-k+1)^2
flatten([[A094250(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Aug 18 2023

Formula

A(n, k) = ((n+2)^(k+1) + (k+1)*n*(n+1) - 1)/(n+1)^2 (array).

T(n, k) = ((n-k+2)^(k+1) + (k+1)*(n-k)*(n-k+1) - 1)/(n-k+1)^2 (antidiagonals).

G.f. for row n: (1-(n+1)*x)/((1-(n+2)*x)*(1-x)^2).