A249010 Expansion of (P(q) - 3*P(q^2) - 5*P(q^5) + 15*P(q^10)) / 8 in powers of q where P() is a Ramanujan Eisenstein series.

1, -3, 0, -12, 6, -3, 0, -24, 18, -39, 0, -36, 24, -42, 0, -12, 42, -54, 0, -60, 6, -96, 0, -72, 72, -3, 0, -120, 48, -90, 0, -96, 90, -144, 0, -24, 78, -114, 0, -168, 18, -126, 0, -132, 72, -39, 0, -144, 168, -171, 0, -216, 84, -162, 0, -36, 144, -240, 0

Offset

0,2

Comments

Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

Links

Table of n, a(n) for n=0..58.

Formula

If n>0 then a(n) = -3 * b(n) where b is multiplicative with b(2^e) = 2 - 2^e, b(5^e) = 1, and b(p^e) = (p^(e+1) - 1) / (p - 1) otherwise.

G.f.: 1 - 3 * Sum_{k>0} c(k) * x^k / (1 - x^k)^2 where c(k) is a period 10 integer sequence.

G.f.: 1 - 3/2 * Sum_{k>0} c(k) * k * x^k / (1 - x^k) where c(k) is a period 10 integer sequence.

a(4*n) = A028887(n). a(4*n + 2) = 0.

Example

G.f. = 1 - 3*q - 12*q^3 + 6*q^4 - 3*q^5 - 24*q^7 + 18*q^8 - 39*q^9 + ...

Prog

(PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); -3 * prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2,  2 - 2^e, if( p==5, 1, (p^(e+1) - 1) / (p - 1))))))};
(PARI) {a(n) = if( n<1, n==0, -3 * sumdiv(n, k, n/k * [8, 1, -2, 1, -2, -4, -2, 1, -2, 1][k%10 + 1]))};
(PARI) {a(n) = if( n<1, n==0, -3/2 * sumdiv(n, k, k * [0, 2, -1, 2, -1, 0, -1, 2, -1, 2][k%10 + 1]))};
(Magma) A := Basis( ModularForms( Gamma0(10), 2), 60); A[1] - 3*A[2];

Crossrefs

Cf. A028887.

Sequence in context: A194093 A055314 A110890 * A071534 A336667 A269880

Adjacent sequences: A249007 A249008 A249009 * A249011 A249012 A249013

Keyword

sign

Author

Michael Somos, Oct 18 2014

Status

approved