%I #41 Jun 28 2024 05:05:06

%S 1,1,1,1,2,1,1,3,2,1,1,4,5,3,1,1,5,10,11,2,1,1,6,17,31,17,4,1,1,7,26,

%T 69,82,39,2,1,1,8,37,131,257,256,65,4,1,1,9,50,223,626,1045,730,139,3,

%U 1,1,10,65,351,1297,3156,4097,2218,261,4,1

%N Square array A(n,k), n >= 1, k >= 0, read by descending antidiagonals, where A(n,k) is Sum_{d|n} k^(d-1).

%H Seiichi Manyama, <a href="/A308813/b308813.txt">Antidiagonals n = 1..140, flattened</a>

%F G.f. of column k: Sum_{j>=1} x^j/(1 - k*x^j).

%F T(n, k) = Sum_{d|(k+1)} (n-k-1)^(d-1), with T(n, n) = 1. - _G. C. Greubel_, Jun 26 2024

%e Square array, A(n,k), begins:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 3, 4, 5, 6, 7, ...

%e 1, 2, 5, 10, 17, 26, 37, ...

%e 1, 3, 11, 31, 69, 131, 223, ...

%e 1, 2, 17, 82, 257, 626, 1297, ...

%e 1, 4, 39, 256, 1045, 3156, 7819, ...

%e 1, 2, 65, 730, 4097, 15626, 46657, ...

%e Antidiagonal triangle, T(n,k), begins as:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 3, 2, 1;

%e 1, 4, 5, 3, 1;

%e 1, 5, 10, 11, 2, 1;

%e 1, 6, 17, 31, 17, 4, 1;

%e 1, 7, 26, 69, 82, 39, 2, 1;

%e 1, 8, 37, 131, 257, 256, 65, 4, 1;

%e 1, 9, 50, 223, 626, 1045, 730, 139, 3, 1;

%e 1, 10, 65, 351, 1297, 3156, 4097, 2218, 261, 4, 1;

%t A[n_, k_] := DivisorSum[n, If[k == # - 1 == 0, 1, k^(# - 1)] &];

%t Table[A[k + 1, n - k - 1], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* _Amiram Eldar_, May 07 2021 *)

%o (Magma)

%o A:= func< n,k | (&+[k^(d-1): d in Divisors(n)]) >;

%o A308813:= func< n,k | A(k+1,n-k-1) >;

%o [A308813(n,k): k in [0..n-1], n in [1..12]]; // _G. C. Greubel_, Jun 26 2024

%o (SageMath)

%o def A(n,k): return sum(k^(j-1) for j in (1..n) if (j).divides(n))

%o def A308813(n,k): return A(k+1,n-k-1)

%o flatten([[A308813(n,k) for k in range(n)] for n in range(1,13)]) # _G. C. Greubel_, Jun 26 2024

%Y Columns k=0..10 give A000012, A000005, A034729, A034730, A339684, A339685, A339686, A339687, A339688, A339689, A113999.

%Y Row n=1..3 give A000012, A000027(n+1), A002522.

%Y A(n,n) gives A308814.

%K nonn,tabl

%O 1,5

%A _Seiichi Manyama_, Jun 26 2019