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

0,3

Links

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

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).

Example

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;

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A122507 A257258 A259325 * A208517 A336203 A209566

Adjacent sequences: A094247 A094248 A094249 * A094251 A094252 A094253

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jun 02 2004

Status

approved