A375877
E.g.f. satisfies A(x) = exp( 3 * (exp(x) - 1) * A(x)^(1/3) ).
Original entry on oeis.org
1, 3, 18, 156, 1785, 25506, 438540, 8834013, 204341580, 5343030264, 155949552951, 5028857184588, 177628447077408, 6822752257361943, 283211285330197254, 12636574861035192648, 603220473535136763441, 30679940004725753797230
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-3*lambertw(1-exp(x)))))
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a(n) = 3*sum(k=0, n, (k+3)^(k-1)*stirling(n, k, 2));
A355788
E.g.f. satisfies log(A(x)) = (exp(2*x) - 1) * A(x)/2.
Original entry on oeis.org
1, 1, 5, 38, 409, 5772, 101227, 2126966, 52153185, 1462998168, 46232500275, 1625693415898, 62972266884721, 2664713395180228, 122315552809623323, 6053803331878334590, 321389617069279569345, 18218906261462603910704, 1098415656103838009681123
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw((1-exp(2*x))/2))))
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a(n) = sum(k=0, n, 2^(n-k)*(k+1)^(k-1)*stirling(n, k, 2));
A356199
a(n) = Sum_{k=0..n} (n*k+1)^(k-1) * Stirling2(n,k).
Original entry on oeis.org
1, 1, 6, 122, 5991, 556152, 84245291, 18956006323, 5940695613628, 2474958812797662, 1323229303771318595, 883245295259143164922, 719968321620942410875645, 703829776430361739799683993, 812798413118207226439408790038, 1094718407894086754989907938078190
Offset: 0
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b:= proc(n, k, m) option remember; `if`(n=0,
(k*m+1)^(m-1), m*b(n-1, k, m)+b(n-1, k, m+1))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..19);
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b[n_, k_, m_] := b[n, k, m] = If[n == 0,
(k*m+1)^(m-1), m*b[n-1, k, m] + b[n-1, k, m+1]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Feb 14 2023, after Alois P. Heinz *)
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a(n) = sum(k=0, n, (n*k+1)^(k-1) * stirling(n, k, 2)); \\ Michel Marcus, Aug 04 2022
A375876
E.g.f. satisfies A(x) = exp( 2 * (exp(x) - 1) * A(x)^(1/2) ).
Original entry on oeis.org
1, 2, 10, 76, 790, 10494, 170396, 3278174, 73019522, 1850066136, 52577005426, 1657084522790, 57382017574920, 2166149552961970, 88550946187572482, 3897682631534087692, 183810990395243463198, 9246950189455617225622, 494332095588897164709644
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2*lambertw(1-exp(x)))))
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a(n) = 2*sum(k=0, n, (k+2)^(k-1)*stirling(n, k, 2));