# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a308813 Showing 1-1 of 1 %I A308813 #41 Jun 28 2024 05:05:06 %S A308813 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 A308813 69,82,39,2,1,1,8,37,131,257,256,65,4,1,1,9,50,223,626,1045,730,139,3, %U A308813 1,1,10,65,351,1297,3156,4097,2218,261,4,1 %N A308813 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 A308813 Seiichi Manyama, Antidiagonals n = 1..140, flattened %F A308813 G.f. of column k: Sum_{j>=1} x^j/(1 - k*x^j). %F A308813 T(n, k) = Sum_{d|(k+1)} (n-k-1)^(d-1), with T(n, n) = 1. - _G. C. Greubel_, Jun 26 2024 %e A308813 Square array, A(n,k), begins: %e A308813 1, 1, 1, 1, 1, 1, 1, ... %e A308813 1, 2, 3, 4, 5, 6, 7, ... %e A308813 1, 2, 5, 10, 17, 26, 37, ... %e A308813 1, 3, 11, 31, 69, 131, 223, ... %e A308813 1, 2, 17, 82, 257, 626, 1297, ... %e A308813 1, 4, 39, 256, 1045, 3156, 7819, ... %e A308813 1, 2, 65, 730, 4097, 15626, 46657, ... %e A308813 Antidiagonal triangle, T(n,k), begins as: %e A308813 1; %e A308813 1, 1; %e A308813 1, 2, 1; %e A308813 1, 3, 2, 1; %e A308813 1, 4, 5, 3, 1; %e A308813 1, 5, 10, 11, 2, 1; %e A308813 1, 6, 17, 31, 17, 4, 1; %e A308813 1, 7, 26, 69, 82, 39, 2, 1; %e A308813 1, 8, 37, 131, 257, 256, 65, 4, 1; %e A308813 1, 9, 50, 223, 626, 1045, 730, 139, 3, 1; %e A308813 1, 10, 65, 351, 1297, 3156, 4097, 2218, 261, 4, 1; %t A308813 A[n_, k_] := DivisorSum[n, If[k == # - 1 == 0, 1, k^(# - 1)] &]; %t A308813 Table[A[k + 1, n - k - 1], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* _Amiram Eldar_, May 07 2021 *) %o A308813 (Magma) %o A308813 A:= func< n,k | (&+[k^(d-1): d in Divisors(n)]) >; %o A308813 A308813:= func< n,k | A(k+1,n-k-1) >; %o A308813 [A308813(n,k): k in [0..n-1], n in [1..12]]; // _G. C. Greubel_, Jun 26 2024 %o A308813 (SageMath) %o A308813 def A(n,k): return sum(k^(j-1) for j in (1..n) if (j).divides(n)) %o A308813 def A308813(n,k): return A(k+1,n-k-1) %o A308813 flatten([[A308813(n,k) for k in range(n)] for n in range(1,13)]) # _G. C. Greubel_, Jun 26 2024 %Y A308813 Columns k=0..10 give A000012, A000005, A034729, A034730, A339684, A339685, A339686, A339687, A339688, A339689, A113999. %Y A308813 Row n=1..3 give A000012, A000027(n+1), A002522. %Y A308813 A(n,n) gives A308814. %K A308813 nonn,tabl %O A308813 1,5 %A A308813 _Seiichi Manyama_, Jun 26 2019 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE