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%I #4 Sep 08 2022 08:46:10
%S 1,-3,0,-12,6,-3,0,-24,18,-39,0,-36,24,-42,0,-12,42,-54,0,-60,6,-96,0,
%T -72,72,-3,0,-120,48,-90,0,-96,90,-144,0,-24,78,-114,0,-168,18,-126,0,
%U -132,72,-39,0,-144,168,-171,0,-216,84,-162,0,-36,144,-240,0
%N 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.
%C Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).
%F 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.
%F G.f.: 1 - 3 * Sum_{k>0} c(k) * x^k / (1 - x^k)^2 where c(k) is a period 10 integer sequence.
%F 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.
%F a(4*n) = A028887(n). a(4*n + 2) = 0.
%e G.f. = 1 - 3*q - 12*q^3 + 6*q^4 - 3*q^5 - 24*q^7 + 18*q^8 - 39*q^9 + ...
%o (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))))))};
%o (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]))};
%o (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]))};
%o (Magma) A := Basis( ModularForms( Gamma0(10), 2), 60); A[1] - 3*A[2];
%Y Cf. A028887.
%K sign
%O 0,2
%A _Michael Somos_, Oct 18 2014
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