OFFSET
1,3
COMMENTS
Euler transform of a(n) - if( n%4, 0, a(n/2)) is sequence itself with offset 0.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..650
FORMULA
G.f.: A(x) = x exp(A(x) - A(-x^2)/2 + A(x^3)/3 - A(-x^4)/4 + ...). Also A(x) = Sum_{n >= 1} a(n)*x^n = x * Product_{n >= 1} (1+(-x)^n)^((-1)^n*a(n)).
G.f.: x prod_{n>0} (1-x^(4n))^a(2n)/(1-x^n)^a(n).
a(n) ~ c * d^n / n^(3/2), where d = 2.92045137601697174071599643..., c = 0.4299447159290328896620383... . - Vaclav Kotesovec, Aug 25 2014
EXAMPLE
x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 18*x^6 + 43*x^7 + 102*x^8 + 247*x^9 + 617*x^10 + ...
MAPLE
with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1 else (add(d*p(d), d=divisors(n)) +add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n-1))/n fi end end: b:= etr(n-> a(n) -`if`(modp(n, 4)<>0, 0, a(n/2))): a:= n-> b(n-1): seq(a(n), n=1..40); # Alois P. Heinz, Sep 06 2008
MATHEMATICA
s[ n_, k_ ] := s[ n, k ]=a[ n+1-k ]+If[ n<2k, 0, -s[ n-k, k ](-1)^k ]; a[ 1 ]=1; a[ n_ ] := a[ n ]=Sum[ a[ i ]s[ n-1, i ]i, {i, 1, n-1} ]/(n-1); Table[ a[ i ], {i, 1, 30} ]
PROG
(PARI) {a(n) = local(A=x); if( n<1, 0, for( k=1, n-1, A *= (1 + (-x)^k + x*O(x^n))^((-1)^k * polcoeff(A, k))); polcoeff(A, n))}
CROSSREFS
KEYWORD
nonn,eigen
AUTHOR
Michael Somos, Aug 08 1999
STATUS
approved