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A349504
E.g.f. satisfies: A(x) = (1 + x)^(A(x)^2).
6
1, 1, 4, 36, 484, 8840, 203868, 5691308, 186612592, 7031373264, 299397454080, 14218443479328, 745142534904480, 42717896158340832, 2659373970144454080, 178666030775042040000, 12884568940594969258752, 992750028716940749121792
OFFSET
0,3
LINKS
FORMULA
a(n) = Sum_{k=0..n} (2*k+1)^(k-1) * Stirling1(n,k).
E.g.f. A(x) = Sum_{k>=0} (2*k+1)^(k-1) * (log(1+x))^k / k!.
From Vaclav Kotesovec, Nov 20 2021: (Start)
E.g.f.: sqrt(-LambertW(-2*log(1 + x)) / (2*log(1 + x))).
a(n) ~ n^(n-1) / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n - 1/2) * exp(n + exp(-1)/4 - 1)). (End)
MATHEMATICA
nmax = 20; CoefficientList[Series[Sqrt[-LambertW[-2*Log[1 + x]]/(2*Log[1 + x])], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 20 2021 *)
PROG
(PARI) a(n) = sum(k=0, n, (2*k+1)^(k-1)*stirling(n, k, 1));
(PARI) N=20; x='x+O('x^N); Vec(serlaplace(sum(k=0, N, (2*k+1)^(k-1)*log(1+x)^k/k!)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Nov 20 2021
STATUS
approved