OFFSET
1,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..400
FORMULA
a(n) ~ c * d^n / n^(3/2), where d = 3.387829329026... and c = 0.51348069856... - Vaclav Kotesovec, Oct 29 2024
EXAMPLE
G.f.: A(x) = x + x^2 + 3*x^3 + 7*x^4 + 20*x^5 + 49*x^6 + 148*x^7 + 391*x^8 + ...
where
log(A(x)/x) = A(x) + (A(x^2) + A(2*x^2))/2 + (A(x^3) + A(3*x^3))/3 + (A(x^4) + A(2*x^4) + A(4*x^4))/4 + (A(x^5) + A(5*x^5))/5 + (A(x^6) + A(2*x^6) + A(3*x^6) + A(6*x^6))/6 + (A(x^7) + A(7*x^7))/7 + (A(x^8) + A(2*x^8) + A(4*x^8) + A(8*x^8))/8 + (A(x^9) + A(3*x^9) + A(9*x^9))/9 + ...
MATHEMATICA
a[n_] := Module[{A=x}, For[i=1, i <= n, i++, A=x*Exp[Sum[DivisorSum[k, (A /. x :> #*x^k + x*O[x]^n)&]/k, {k, 1, n}]]]; SeriesCoefficient[A, n]]; Table[Print[an = a[n]]; an, {n, 1, 35}] (* Jean-François Alcover, Dec 10 2015, adapted from PARI *)
PROG
(PARI) {a(n)=local(A=x); for(i=1, n, A=x*exp(sum(k=1, n, sumdiv(k, d, subst(A, x, d*x^k +x*O(x^n)))/k))); polcoeff(A, n)}
for(n=1, 35, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 16 2013
STATUS
approved