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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. 4
 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, 901, 17713, 13729, 1, 0, 1, 1, 31, 2861, 249337, 4572529, 943227, 1, 0, 1, 1, 43, 7291, 1900521, 264273961, 3426693463, 126433847, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 COMMENTS A(n,k) is odd if k >= 1 or n = 0. LINKS Alois P. Heinz, Antidiagonals n = 0..43, flattened Richard Stanley, Proof of the general conjecture, MathOverflow, March 2021. FORMULA E.g.f. of column k>0: (Sum_{j>=0} k^(j*(j+1)/2) * x^j/j!)^(1/k). E.g.f. of column k=0: 1. A(n,k) == 1 (mod k*(k-1)) for k >= 2 (see "general conjecture" in A178319 and link to proof by Richard Stanley above). EXAMPLE Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, ... 0, 1, 1, 1, 1, 1, ... 0, 1, 3, 7, 13, 21, ... 0, 1, 23, 199, 901, 2861, ... 0, 1, 393, 17713, 249337, 1900521, ... 0, 1, 13729, 4572529, 264273961, 6062674201, ... ... MAPLE A:= (n, k)-> `if`(k>0, n!*coeff(series(add(k^(j*(j+1)/2)* x^j/j!, j=0..n)^(1/k), x, n+1), x, n), k^n): seq(seq(A(n, d-n), n=0..d), d=0..10); CROSSREFS Columns k=0-3 give: A000007, A000012, A178315, A178319. Rows n=0-2 give: A000012, A057427, A002061 (for k>0). Main diagonal gives A342578. Cf. A000217, A023813. Sequence in context: A327622 A183134 A328747 * A053382 A031253 A291624 Adjacent sequences: A346058 A346059 A346060 * A346062 A346063 A346064 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jul 03 2021 STATUS approved

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Last modified August 11 23:45 EDT 2024. Contains 375082 sequences. (Running on oeis4.)