A200402 G.f. satisfies: A(x) = exp( Sum_{n>=1} A(-x^n)^3 * x^n/n ).
1, 1, -2, -5, 24, 81, -439, -1590, 9144, 34451, -206641, -799196, 4936378, 19442800, -122613798, -489411508, 3134773097, 12640278932, -81948641010, -333099985517, 2180523864984, 8920922434686, -58861487584914, -242105281357185, 1608002839956522, 6643707274089977, -44372373955131024
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x - 2*x^2 - 5*x^3 + 24*x^4 + 81*x^5 - 439*x^6 +... where log(A(x)) = A(-x)^3*x + A(-x^2)^3*x^2/2 + A(-x^3)^3*x^3/3 + A(-x^4)^3*x^4/4 +... The coefficients in A(-x)^3 begin: [1,-3,-3,26,48,-444,-920,9126,19587,-204214,-449496,4841001,...] and the g.f. may be expressed by the Euler product: A(x) = 1/((1-x)^1*(1-x^2)^-3*(1-x^3)^-3*(1-x^4)^26*(1-x^5)^48*(1-x^6)^-444*(1-x^7)^-920*(1-x^8)^9126*...).
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1347
Programs
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Maple
b:= proc(n) option remember; add(a(i)*a(n-i), i=0..n) end: g:= proc(n) option remember; (-1)^n*add(b(i)*a(n-i), i=0..n) end: a:= proc(n) option remember; `if`(n=0, 1, add(add( d*g(d-1), d=numtheory[divisors](j))*a(n-j), j=1..n)/n) end: seq(a(n), n=0..30); # Alois P. Heinz, Jan 24 2017 -
Mathematica
b[n_] := b[n] = Sum[a[i]*a[n-i], {i, 0, n}]; g[n_] := g[n] = (-1)^n*Sum[b[i]*a[n-i], {i, 0, n}]; a[n_] := a[n] = If[n==0, 1, Sum[DivisorSum[j, #*g[#-1]&]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 24 2017, after Alois P. Heinz *) -
PARI
{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,subst(A^3,x,-x^m)*x^m/m)+x*O(x^n)));polcoeff(A,n)}
Formula
Equals the Euler transformation of the coefficients in A(-x)^3, where A(x) is the g.f. of this sequence.
Comments