OFFSET
0,4
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..77
Kimmo Berg, Complexity of solution structures in nonlinear pricing, Ann. Oper. Res. 206, 23-37 (2013).
Sergei Chmutov, Maxim Kazarian and 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)
a(n) = (-1)^(n-1)*A003025(n)/n. - Andrew Howroyd, Jan 07 2022
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
KEYWORD
sign,easy
AUTHOR
Paul D. Hanna, Oct 30 2007
STATUS
approved