A047848 - OEIS

%I #15 Jan 14 2025 10:30:20

%S 1,2,1,5,2,1,14,6,2,1,41,22,7,2,1,122,86,32,8,2,1,365,342,157,44,9,2,

%T 1,1094,1366,782,260,58,10,2,1,3281,5462,3907,1556,401,74,11,2,1,9842,

%U 21846,19532,9332,2802,586,92,12,2,1,29525,87382,97657,55988,19609,4682,821,112,13,2,1

%N Array A read by diagonals; n-th difference of (A(k,n), A(k,n-1),..., A(k,0)) is (k+2)^(n-1), for n=1,2,3,...; k=0,1,2,...

%H G. C. Greubel, <a href="/A047848/b047848.txt">Antidiagonals n = 0..50, flattened</a>

%F A(n, k) = ((n+3)^k + n + 1)/(n+2). - _Ralf Stephan_, Feb 14 2004

%F From _G. C. Greubel_, Jan 11 2025: (Start)

%F T(n, k) = ((k+3)^(n-k) + k + 1)/(k+2) (antidiagonal triangle).

%F T(n, n) = A196793(n).

%F Sum_{k=0..n} T(n, k) = A047857(n). (End)

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

%e   1, 2,  5,  14,   41, ... = A007051.

%e   1, 2,  6,  22,   86, ... = A047849.

%e   1, 2,  7,  32,  157, ... = A047850.

%e   1, 2,  8,  44,  260, ... = A047851.

%e   1, 2,  9,  58,  401, ... = A047852.

%e   1, 2, 10,  74,  586, ... = A047853.

%e   1, 2, 11,  92,  821, ... = A047854.

%e   1, 2, 12, 112, 1112, ... = A047855.

%e   1, 2, 13, 134, 1465, ... = A047856.

%e   1, 2, 14, 158, 1886, ... = A196791.

%e   1, 2, 15, 184, 2381, ... = A196792.

%e Downward antidiagonals, T(n, k), begins as:

%e       1;

%e       2,     1;

%e       5,     2,     1;

%e      14,     6,     2,     1;

%e      41,    22,     7,     2,     1;

%e     122,    86,    32,     8,     2,    1;

%e     365,   342,   157,    44,     9,    2,   1;

%e    1094,  1366,   782,   260,    58,   10,   2,   1;

%e    3281,  5462,  3907,  1556,   401,   74,  11,   2,  1;

%e    9842, 21846, 19532,  9332,  2802,  586,  92,  12,  2, 1;

%e   29525, 87382, 97657, 55988, 19609, 4682, 821, 112, 13, 2, 1;

%t A[n_, k_]:= ((n+3)^k +n+1)/(n+2);

%t A047848[n_, k_]:= A[k,n-k];

%t Table[A047848[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 11 2025 *)

%o (Magma)

%o A:= func< n,k | ((n+3)^k +n+1)/(n+2) >; // array A047848

%o [A(k,n-k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 11 2025

%o (Python)

%o def A(n,k): return (pow(n+3,k) +n+1)//(n+2) # array A047848

%o print(flatten([[A(k,n-k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Jan 11 2025

%Y Cf. A047857 (row sums), A196793 (main diagonal).

%K nonn,tabl,easy

%O 0,2

%A _Clark Kimberling_