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 A134531 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)] ). 5
 0, 1, -1, 5, -79, 3377, -362431, 93473345, -56272471039, 77442176448257, -239804482525402111, 1650172344732021412865, -24981899010711376986398719, 825164608171793476724052668417, -59053816996641612758331731690504191, 9102696765174239045811746247171452452865 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Sergei Chmutov, Maxim Kazarian, Sergey Lando, Polynomial graph invariants and the KP hierarchy, arXiv:1803.09800 [math.CO], 2018. FORMULA 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). From Peter Bala, Apr 01 2013: (Start) 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) - .... Recurrence equation: a(n) = 1 - sum {k = 1..n-1} 2^(k*(n-k))*C(n-1,k-1)*a(k) with a(1) = 1. (End) EXAMPLE Let g.f. G(x) = Sum_{n>=0} a(n)*x^n/[ n! * 2^(n*(n-1)/2) ] then exp(G(x)) = Sum_{n>=0} x^n/[ n! * 2^(n*(n-1)/2) ]; G.f.: G(x) = x - x^2/4 + 5x^3/48 - 79x^4/1536 + 3377x^5/122880 +... exp(G(x)) = 1 + x + x^2/4 + x^3/48 + x^4/1536 + x^5/122880 +... MATHEMATICA a[0] = 0; a[n_] := a[n] = 1 - Sum[2^(k(n-k)) Binomial[n-1, k-1] a[k], {k, 1, n-1}]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 26 2018 *) PROG (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)} CROSSREFS Cf. related triangles: A134530, A111636; A118197 (variant); A011266. Sequence in context: A244585 A293786 A141828 * A062250 A131284 A105917 Adjacent sequences:  A134528 A134529 A134530 * A134532 A134533 A134534 KEYWORD sign,easy AUTHOR Paul D. Hanna, Oct 30 2007 STATUS approved

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Last modified November 18 13:22 EST 2018. Contains 317306 sequences. (Running on oeis4.)