OFFSET
1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..5000
FORMULA
a(n) is multiplicative with a(p^e) = ((p^4)^(e+1) - s^(e+1)) / (p^4 - s) where s = 0 if p = 3, s = 1 if p == 1 (mod 3), s = -1 if p == 2 (mod 3).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = 4*Pi^5/(729*sqrt(3)) = 0.9694405... (A344778). - Amiram Eldar, Nov 09 2023
EXAMPLE
G.f. = x + 15*x^2 + 81*x^3 + 241*x^4 + 624*x^5 + 1215*x^6 + 2402*x^7 + ...
MATHEMATICA
a[ n_] := If[ n < 2, Boole[n == 1], Times @@ (With[{s = {1, -1, 0}[[Mod[#, 3, 1]]]}, ((#^4)^(#2 + 1) - s^(#2 + 1)) / (#^4 - s)] & @@@ FactorInteger[n])];
PROG
(PARI) {a(n) = my(A, U1, u3, U9); if( n<1, 0, n--; A = x * O(x^n); U1 = eta(x + A)^3; u3 = eta(x^3 + A); U9 = eta(x^9 + A)^3; polcoeff( U1 * u3^7 * (1 + 9*x*U9/U1)^2, n))};
(PARI) {a(n) = my(A, p, e, s); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, p^(4*e), s=-(-1)^(p%3); ((p^4)^(e+1) - s^(e+1)) / (p^4 - s))))};
(Magma) A := Basis( ModularForms( Gamma1(3), 5), 37); A[2];
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Michael Somos, Dec 26 2015
STATUS
approved