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%I #21 Feb 08 2022 23:22:26
%S 1,3,9,19,42,78,146,249,429,695,1125,1749,2713,4086,6123,8986,13122,
%T 18852,26934,38001,53328,74068,102336,140208,191153,258741,348606,
%U 466806,622383,825342,1090087,1432923,1876542,2447029,3179859,4116282,5311204,6829008
%N McKay-Thompson series of class 15C for the Monster group with a(0) = 3.
%C Equals the triangular series convolved with the aerated variant of A153084; i.e., (1, 3, 6, 10, ...) * (1, 0, 3, 0, 9, ...). Note that (1, 3, 6, 10, ...) convolved with (1, 0, 3, 0, 6, ...) = A038163: (1, 3, 9, 19, 39, 69, ...). - _Gary W. Adamson_, Aug 12 2016
%H G. A. Edgar, <a href="/A153084/b153084.txt">Table of n, a(n) for n = -1..997</a>
%F Expansion of (eta(q^3) * eta(q^5) / (eta(q) * eta(q^15)))^3 in powers of q.
%F Euler transform of period 15 sequence [ 3, 3, 0, 3, 0, 0, 3, 3, 0, 0, 3, 0, 3, 3, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + v) * (u^2 + 5*u*v + v^2) - u*v * (u*v - 1).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = f(t) where q = exp(2 Pi i t).
%F G.f. x^(-1) * (Prod_{k>0} ( (1 - x^(3*k)) * (1 - x^(5*k)) / ((1 - x^k) * (1 - x^(15*k)))))^3. [corrected by _Vaclav Kotesovec_, Jun 26 2018]
%F a(n) ~ exp(4*Pi*sqrt(n/15)) / (sqrt(2) * 15^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 26 2018
%e T15C = 1/q + 3 + 9*q + 19*q^2 + 42*q^3 + 78*q^4 + 146*q^5 + 249*q^6 + 429*q^7 + ...
%e 42 = (1, 3, 6, 10, 15) dot (9, 0, 3, 0, 1) = (9 + 0 + 18 + 0 + 15). - _Gary W. Adamson_, Aug 12 2016
%t nmax=60; CoefficientList[Series[Product[(((1-x^(3*k)) * (1-x^(5*k)) / (1-x^(15*k)) / (1-x^k)))^3,{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 13 2015, typo corrected by _Vaclav Kotesovec_, Jun 26 2018 *)
%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^5 + A) / (eta(x + A) * eta(x^15 + A)))^3, n))}
%Y A058510(n) = a(n) unless n=0. Convolution inverse of A095123.
%Y Cf. A038163.
%K nonn
%O -1,2
%A _Michael Somos_, Dec 18 2008
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