|
%I #30 Sep 04 2022 17:10:23
%S 1,1,-1,1,-1,-1,1,-1,-2,0,1,-1,-4,-1,0,1,-1,-8,-5,-1,1,1,-1,-16,-19,
%T -7,5,0,1,-1,-32,-65,-37,27,1,1,1,-1,-64,-211,-175,155,17,13,0,1,-1,
%U -128,-665,-781,927,205,167,4,0,1,-1,-256,-2059,-3367,5675,2129,2089,110,0,0
%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j^k*x^j).
%H Seiichi Manyama, <a href="/A292166/b292166.txt">Antidiagonals n = 0..139, flattened</a>
%F A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(1+k*j/d)) * A(n-j,k) for n > 0. - _Seiichi Manyama_, Nov 02 2017
%e Square array begins:
%e 1, 1, 1, 1, 1, ...
%e -1, -1, -1, -1, -1, ...
%e -1, -2, -4, -8, -16, ...
%e 0, -1, -5, -19, -65, ...
%e 0, -1, -7, -37, -175, ...
%t A[n_, k_] := A[n, k] = If[n == 0, 1, -(1/n)*Sum[Sum[d^(1+k*j/d), {d, Divisors[j]}]*A[n-j, k], {j, 1, n}]];
%t Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Sep 04 2022 *)
%Y Columns k=0..2 give A010815, A022661, A292164.
%Y Rows n=0..2 give A000012, (-1)*A000012, (-1)*A000079.
%Y Main diagonal gives A292167.
%K sign,look,tabl
%O 0,9
%A _Seiichi Manyama_, Sep 10 2017
|
