A094250 - OEIS

%I #9 Aug 20 2023 10:50:08

%S 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,

%T 127,1,3,12,55,199,459,550,255,1,3,13,70,315,981,1825,1644,511,1,3,14,

%U 87,471,1871,4888,7287,4925,1023,1,3,15,106,673,3273,11203,24420,29133,14767,2047

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

%H G. C. Greubel, <a href="/A094250/b094250.txt">Antidiagonals n = 0..50, flattened</a>

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

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

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

%e Array, A(n, k), begins:

%e   1, 3,  7, 15,  31,   63,   127,    255,     511, ... A000225;

%e   1, 3,  8, 22,  63,  185,   550,   1644,    4925, ... A047926;

%e   1, 3,  9, 31, 117,  459,  1825,   7287,   29133, ... A073724;

%e   1, 3, 10, 42, 199,  981,  4888,  24420,  122077, ... A094195;

%e   1, 3, 11, 55, 315, 1871, 11203,  67191,  403115, ... A094259;

%e   1, 3, 12, 70, 471, 3273, 22882, 160140, 1120941, ...

%e Antidiagonals, T(n, k), begins as:

%e   1;

%e   1, 3;

%e   1, 3,  7;

%e   1, 3,  8, 15;

%e   1, 3,  9, 22,  31;

%e   1, 3, 10, 31,  63,   63;

%e   1, 3, 11, 42, 117,  185,  127;

%e   1, 3, 12, 55, 199,  459,  550,  255;

%e   1, 3, 13, 70, 315,  981, 1825, 1644,  511;

%e   1, 3, 14, 87, 471, 1871, 4888, 7287, 4925, 1023;

%t A094250[n_, k_]:= ((n-k+2)^(k+1) + (k+1)*(n-k)*(n-k+1) - 1)/(n-k+1)^2;

%t Table[A094250[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Aug 18 2023 *)

%o (Magma)

%o A094250:= func< n,k | ((n-k+2)^(k+1) + (k+1)*(n-k)*(n-k+1) - 1)/(n-k+1)^2 >;

%o [A094250(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Aug 18 2023

%o (SageMath)

%o def A094250(n, k): return ((n-k+2)^(k+1) + (k+1)*(n-k)*(n-k+1) - 1)/(n-k+1)^2

%o flatten([[A094250(n,k) for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, Aug 18 2023

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

%K nonn,tabl

%O 0,3

%A _N. J. A. Sloane_, Jun 02 2004