cp's OEIS Frontend

A383033 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Weigh transform of j-> k^(j-1).

%I A383033 #11 Apr 13 2025 11:41:12
%S A383033 1,1,1,1,2,0,1,3,2,1,1,4,6,5,0,1,5,12,18,6,0,1,6,20,46,42,11,0,1,7,30,
%T A383033 95,156,113,18,1,1,8,42,171,420,566,294,35,0,1,9,56,280,930,1930,2028,
%U A383033 798,56,0,1,10,72,428,1806,5185,8820,7396,2128,105,0
%N A383033 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Weigh transform of j-> k^(j-1).
%H A383033 Christian G. Bower, <a href="https://oeis.org/transforms_pari.txt">PARI programs for transforms</a>, 2007.
%H A383033 N. J. A. Sloane, <a href="/transforms.txt">Maple programs for transforms</a>, 2001-2020.
%F A383033 A(n,k) = (1/n) * (k^n - (k-1)^n + Sum_{d<n and d|n} (-1)^(n/d) * d * A(d,k)).
%F A383033 A(n,k) = A383023(n,k) - A383023(n,k-1).
%F A383033 Product_{n>=1} (1 + x^n)^A(n,k) = (1 - (k-1)*x)/(1 - k*x).
%e A383033 Square array begins:
%e A383033   1,  1,   1,    1,    1,     1,     1, ...
%e A383033   1,  2,   3,    4,    5,     6,     7, ...
%e A383033   0,  2,   6,   12,   20,    30,    42, ...
%e A383033   1,  5,  18,   46,   95,   171,   280, ...
%e A383033   0,  6,  42,  156,  420,   930,  1806, ...
%e A383033   0, 11, 113,  566, 1930,  5185, 11851, ...
%e A383033   0, 18, 294, 2028, 8820, 28830, 77658, ...
%Y A383033 Columns k=1..3 give A209229, A383034, A383035.
%Y A383033 Main diagonal gives A316073.
%Y A383033 Cf. A383023.
%K A383033 nonn,tabl
%O A383033 1,5
%A A383033 _Seiichi Manyama_, Apr 13 2025