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A346061 A(n,k) = n! * [x^n] (Sum_{j=0..n} k^(j*(j+1)/2) * x^j/j!)^(1/k) if k>0, A(n,0) = 0^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A346061 #25 Jul 05 2021 12:02:50
%S A346061 1,1,0,1,1,0,1,1,1,0,1,1,3,1,0,1,1,7,23,1,0,1,1,13,199,393,1,0,1,1,21,
%T A346061 901,17713,13729,1,0,1,1,31,2861,249337,4572529,943227,1,0,1,1,43,
%U A346061 7291,1900521,264273961,3426693463,126433847,1,0
%N A346061 A(n,k) = n! * [x^n] (Sum_{j=0..n} k^(j*(j+1)/2) * x^j/j!)^(1/k) if k>0, A(n,0) = 0^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A346061 A(n,k) is odd if k >= 1 or n = 0.
%H A346061 Alois P. Heinz, <a href="/A346061/b346061.txt">Antidiagonals n = 0..43, flattened</a>
%H A346061 Richard Stanley, <a href="https://mathoverflow.net/q/385402">Proof of the general conjecture</a>, MathOverflow, March 2021.
%F A346061 E.g.f. of column k>0: (Sum_{j>=0} k^(j*(j+1)/2) * x^j/j!)^(1/k).
%F A346061 E.g.f. of column k=0: 1.
%F A346061 A(n,k) == 1 (mod k*(k-1)) for k >= 2 (see "general conjecture" in A178319 and link to proof by _Richard Stanley_ above).
%e A346061 Square array A(n,k) begins:
%e A346061   1, 1,     1,       1,         1,          1, ...
%e A346061   0, 1,     1,       1,         1,          1, ...
%e A346061   0, 1,     3,       7,        13,         21, ...
%e A346061   0, 1,    23,     199,       901,       2861, ...
%e A346061   0, 1,   393,   17713,    249337,    1900521, ...
%e A346061   0, 1, 13729, 4572529, 264273961, 6062674201, ...
%e A346061   ...
%p A346061 A:= (n, k)-> `if`(k>0, n!*coeff(series(add(k^(j*(j+1)/2)*
%p A346061              x^j/j!, j=0..n)^(1/k), x, n+1), x, n), k^n):
%p A346061 seq(seq(A(n, d-n), n=0..d), d=0..10);
%Y A346061 Columns k=0-3 give: A000007, A000012, A178315, A178319.
%Y A346061 Rows n=0-2 give: A000012, A057427, A002061 (for k>0).
%Y A346061 Main diagonal gives A342578.
%Y A346061 Cf. A000217, A023813.
%K A346061 nonn,tabl
%O A346061 0,13
%A A346061 _Alois P. Heinz_, Jul 03 2021