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%I #27 Jun 18 2022 10:28:54
%S 0,1,-1,5,-79,3377,-362431,93473345,-56272471039,77442176448257,
%T -239804482525402111,1650172344732021412865,
%U -24981899010711376986398719,825164608171793476724052668417,-59053816996641612758331731690504191,9102696765174239045811746247171452452865
%N G.f.: Sum_{n>=0} a(n)*x^n/(n!*2^(n*(n-1)/2)) = log( Sum_{n>=0} x^n/(n!*2^(n*(n-1)/2)) ).
%H Seiichi Manyama, <a href="/A134531/b134531.txt">Table of n, a(n) for n = 0..77</a>
%H Kimmo Berg, <a href="https://doi.org/10.1007/s10479-013-1334-3">Complexity of solution structures in nonlinear pricing</a>, Ann. Oper. Res. 206, 23-37 (2013).
%H Sergei Chmutov, Maxim Kazarian and Sergey Lando, <a href="https://arxiv.org/abs/1803.09800">Polynomial graph invariants and the KP hierarchy</a>, arXiv:1803.09800 [math.CO], 2018.
%F Equals column 0 of triangle A134530, which is the matrix log of triangle A111636, where A111636(n,k) = (2^k)^(n-k)*C(n,k).
%F From _Peter Bala_, Apr 01 2013: (Start)
%F Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)). Then a generating function for this sequence (but with a different offset) is E(x)/E(2*x) = Sum_{n >= 0} a(n-1)*x^n/(n!*2^C(n,2)) = 1 - x + 5*x^2/(2!*2) - 79*x^3/(3!*2^3) + 3377*x^4/(4!*2^6) - ....
%F Recurrence equation:
%F a(n) = 1 - Sum_{k = 1..n-1} 2^(k*(n-k))*C(n-1,k-1)*a(k) with a(1) = 1. (End)
%F a(n) = (-1)^(n-1)*A003025(n)/n. - _Andrew Howroyd_, Jan 07 2022
%e Let g.f. G(x) = Sum_{n>=0} a(n)*x^n/[ n! * 2^(n*(n-1)/2) ]
%e then exp(G(x)) = Sum_{n>=0} x^n/[ n! * 2^(n*(n-1)/2) ];
%e G.f.: G(x) = x - x^2/4 + 5x^3/48 - 79x^4/1536 + 3377x^5/122880 + ...
%e exp(G(x)) = 1 + x + x^2/4 + x^3/48 + x^4/1536 + x^5/122880 + ...
%t a[0] = 0;
%t a[n_] := a[n] = 1 - Sum[2^(k(n-k)) Binomial[n-1, k-1] a[k], {k, 1, n-1}];
%t Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Jul 26 2018 *)
%o (PARI) {a(n)=n!*2^(n*(n-1)/2)*polcoeff(log(sum(k=0,n,x^k/(k!*2^(k*(k-1)/2)))+x*O(x^n)),n)}
%Y Cf. related triangles: A134530, A111636.
%Y Cf. A003025, A011266, A118197 (variant).
%K sign,easy
%O 0,4
%A _Paul D. Hanna_, Oct 30 2007
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