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%I #30 Apr 19 2024 07:25:59
%S 0,1,0,1,1,8,26,194,1142,9736,81384,823392,8738016,104336880,
%T 1328270880,18419317968,272291315376,4312675967232,72478365279360,
%U 1292173575000192,24314102888206464,482046102448383744,10037081891973037824
%N Expansion of e.g.f. log(1-log(1-x)).
%C Stirling transform of a(n)=[1,0,1,1,8,26,...] is A075792(n)=[1,1,2,8,44,...]. - _Michael Somos_, Mar 04 2004
%C 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
%D G. H. Hardy, A Course of Pure Mathematics, 10th ed., Cambridge University Press, 1960, p. 428.
%H Seiichi Manyama, <a href="/A089064/b089064.txt">Table of n, a(n) for n = 0..451</a>
%H G. H. Hardy, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=ACM1516.0001.001">A Course of Pure Mathematics</a>, Cambridge, The University Press, 1908.
%F a(n) = (-1)^(n+1)*Sum_{k=1..n} (k-1)!*Stirling1(n, k).
%F E.g.f.: log(1-log(1-x)).
%F a(n) = (n-1)! - Sum_{k=1..n-1} binomial(n-1,k) * (k-1)! * a(n-k). - _Seiichi Manyama_, Jun 01 2019
%t nmax = 20; CoefficientList[Series[Log[1-Log[1-x]], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jul 01 2018 *)
%t Table[(-1)^(n+1) * Sum[(k-1)! * StirlingS1[n, k], {k, 1, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 19 2024 *)
%o (PARI) a(n)=if(n<0,0,n!*polcoeff(log(1-log(1-x+x*O(x^n))),n))
%o (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
%Y Cf. A003713, A075792.
%K easy,nonn,changed
%O 0,6
%A _Vladeta Jovovic_, Dec 20 2003
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