A349588 - OEIS

%I #17 Nov 25 2021 08:52:17

%S 1,1,2,8,47,367,3592,42317,583522,9223872,164482761,3267077365,

%T 71540314562,1712334954865,44479256704898,1246241906483516,

%U 37465750470667023,1202986323660907447,41089436549405467096,1487622596267089224901,56907111260864275384346

%N E.g.f. satisfies: A(x) * log(A(x)) = exp(x*A(x)) - 1.

%H Seiichi Manyama, <a href="/A349588/b349588.txt">Table of n, a(n) for n = 0..403</a>

%F a(n) = Sum_{k=0..n} (n-k+1)^(k-1) * Stirling2(n,k).

%F a(n) ~ s * n^(n-1) / (sqrt(r*s - 1/(1 + log(s))) * exp(n) * r^n), where r = 0.4858893246242883887847088396703818017675758048583... and s = 3.016426175038226058288579473351450432292607021364... are roots of the system of equations exp(r*s) = 1 + s*log(s), exp(r*s)*r = 1 + log(s). - _Vaclav Kotesovec_, Nov 25 2021

%t a[n_] := Sum[(n - k + 1)^(k - 1)*StirlingS2[n, k], {k, 0, n}]; Array[a, 21, 0] (* _Amiram Eldar_, Nov 23 2021 *)

%o (PARI) a(n) = sum(k=0, n, (n-k+1)^(k-1)*stirling(n, k, 2));

%Y Cf. A141209, A349587, A349589.

%Y Cf. A349557.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 22 2021