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A383042 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 Euler transform of j-> k^(j-1).

%I A383042 #18 Apr 14 2025 07:40:09
%S A383042 1,1,0,1,1,0,1,2,2,0,1,3,6,3,0,1,4,12,15,6,0,1,5,20,42,42,9,0,1,6,30,
%T A383042 90,156,107,18,0,1,7,42,165,420,554,294,30,0,1,8,56,273,930,1910,2028,
%U A383042 780,56,0,1,9,72,420,1806,5155,8820,7350,2128,99,0
%N A383042 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 Euler transform of j-> k^(j-1).
%H A383042 Christian G. Bower, <a href="https://oeis.org/transforms_pari.txt">PARI programs for transforms</a>, 2007.
%H A383042 N. J. A. Sloane, <a href="/transforms.txt">Maple programs for transforms</a>, 2001-2020.
%F A383042 A(n,k) = (1/n) * Sum_{d|n} mu(n/d) * (k^d - (k-1)^d).
%F A383042 A(n,k) = (1/n) * (k^n - (k-1)^n - Sum_{d<n and d|n} d * A(d,k)).
%F A383042 A(n,k) = A074650(n,k) - A074650(n,k-1).
%F A383042 Product_{n>=1} 1/(1 - x^n)^A(n,k) = (1 - (k-1)*x)/(1 - k*x).
%F A383042 G.f. of column k: Sum_{j>=1} mu(j) * log(1 + x^j/(1 - k*x^j)) / j.
%e A383042 Square array begins:
%e A383042   1,  1,   1,    1,    1,     1,     1, ...
%e A383042   0,  1,   2,    3,    4,     5,     6, ...
%e A383042   0,  2,   6,   12,   20,    30,    42, ...
%e A383042   0,  3,  15,   42,   90,   165,   273, ...
%e A383042   0,  6,  42,  156,  420,   930,  1806, ...
%e A383042   0,  9, 107,  554, 1910,  5155, 11809, ...
%e A383042   0, 18, 294, 2028, 8820, 28830, 77658, ...
%e A383042   ...
%o A383042 (PARI) a(n, k) = sumdiv(n, d, moebius(n/d)*(k^d-(k-1)^d))/n;
%Y A383042 Columns k=1..5 give A000007, A059966, A065178, A065179, A065180.
%Y A383042 Main diagonal gives A306173.
%Y A383042 Cf. A065177 (another version).
%Y A383042 Cf. A008683, A074650, A383033.
%K A383042 nonn,tabl
%O A383042 1,8
%A A383042 _Seiichi Manyama_, Apr 13 2025