A089064 Expansion of e.g.f. log(1-log(1-x)).

0, 1, 0, 1, 1, 8, 26, 194, 1142, 9736, 81384, 823392, 8738016, 104336880, 1328270880, 18419317968, 272291315376, 4312675967232, 72478365279360, 1292173575000192, 24314102888206464, 482046102448383744, 10037081891973037824

Offset: 0

Author

Vladeta Jovovic, Dec 20 2003

Keywords

• easy
• nonn

Comments

Stirling transform of a(n)=[1,0,1,1,8,26,...] is A075792(n)=[1,1,2,8,44,...]. - Michael Somos, Mar 04 2004

Stirling transform of -(-1)^n*a(n)=[1,0,1,-1,8,-26,194,...] is A000142(n-1)=[1,1,2,6,24,120,...]. - Michael Somos, Mar 04 2004

Number of increasing trees on n vertices in which the second player has a winning strategy when interpreted as a game tree - Victor YZ Wang, May 03 2025

Convolution of absolute value of Mobius function and Mobius function applied to bottom and top elements of the set partition lattice - Victor YZ Wang, May 03 2025

References

• G. H. Hardy, A Course of Pure Mathematics, 10th ed., Cambridge University Press, 1960, p. 428.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..451
• G. H. Hardy, A Course of Pure Mathematics, Cambridge, The University Press, 1908.
• Victor Wang On an alternating sum of factorials and Stirling numbers of the first kind: trees, lattices, and games, arXiv:2504.21176 [math.CO], 2025.

Crossrefs

Cf. A003713, A075792.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Log[1-Log[1-x]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 01 2018 *)
Table[(-1)^(n+1) * Sum[(k-1)! * StirlingS1[n, k], {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 19 2024 *)

PARI

a(n)=if(n<0,0,n!*polcoeff(log(1-log(1-x+x*O(x^n))),n))

PARI

{a(n) = if (n<1, 0, (n-1)!-sum(k=1, n-1, binomial(n-1, k)*(k-1)!*a(n-k)))} \\ Seiichi Manyama, Jun 01 2019

Formula

a(n) = (-1)^(n+1)*Sum_{k=1..n} (k-1)!*Stirling1(n, k).

E.g.f.: log(1-log(1-x)).

a(n) = (n-1)! - Sum_{k=1..n-1} binomial(n-1,k) * (k-1)! * a(n-k). - Seiichi Manyama, Jun 01 2019