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A383023 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.

%I A383023 #14 Apr 13 2025 08:19:10
%S A383023 1,2,1,3,3,0,4,6,2,1,5,10,8,6,0,6,15,20,24,6,0,7,21,40,70,48,11,0,8,
%T A383023 28,70,165,204,124,18,1,9,36,112,336,624,690,312,36,0,10,45,168,616,
%U A383023 1554,2620,2340,834,56,0,11,55,240,1044,3360,7805,11160,8230,2184,105,0
%N A383023 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.
%H A383023 Christian G. Bower, <a href="https://oeis.org/transforms_pari.txt">PARI programs for transforms</a>, 2007.
%H A383023 N. J. A. Sloane, <a href="/transforms.txt">Maple programs for transforms</a>, 2001-2020.
%F A383023 A(n,k) = (1/n) * (k^n + Sum_{d<n and d|n} (-1)^(n/d) * d * A(d,k)).
%F A383023 Product_{n>=1} (1 + x^n)^A(n,k) = 1/(1 - k*x).
%e A383023 Square array begins:
%e A383023   1,  2,   3,    4,     5,     6,      7, ...
%e A383023   1,  3,   6,   10,    15,    21,     28, ...
%e A383023   0,  2,   8,   20,    40,    70,    112, ...
%e A383023   1,  6,  24,   70,   165,   336,    616, ...
%e A383023   0,  6,  48,  204,   624,  1554,   3360, ...
%e A383023   0, 11, 124,  690,  2620,  7805,  19656, ...
%e A383023   0, 18, 312, 2340, 11160, 39990, 117648, ...
%Y A383023 Columns k=1..5 give A209229, A306156, A306157, A306158, A306159.
%Y A383023 Cf. A074650.
%K A383023 nonn,tabl
%O A383023 1,2
%A A383023 _Seiichi Manyama_, Apr 12 2025