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%I #26 Oct 06 2019 19:26:13
%S 1,1,1,2,2,3,4,5,6,7,9,11,13,16,19,23,27,32,38,44,52,61,71,82,95,110,
%T 127,145,167,191,218,249,283,322,365,414,469,529,597,673,757,851,955,
%U 1071,1199,1341,1499,1673,1865,2078,2313,2572,2857,3171,3517,3897
%N Coefficients of replicable function number "54b".
%C a(n) is the number of partitions of n into distinct parts where no part is a multiple of 9. - _Joerg Arndt_, Aug 31 2015
%C In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^k)/(1 + x^(m*k)), then a(n) ~ exp(Pi*sqrt((m-1)*n/(3*m))) * (m-1)^(1/4) / (2^(3/2) * 3^(1/4) * m^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Aug 31 2015
%H Alois P. Heinz, <a href="/A112193/b112193.txt">Table of n, a(n) for n = 0..10000</a>
%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Comm. Algebra 22, No. 13, 5175-5193 (1994).
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F a(n) ~ exp(2*Pi*sqrt(2*n/3)/3) / (6^(3/4) * n^(3/4)) * (1 - (9*sqrt(3)/ (16*Pi*sqrt(2)) + sqrt(2)*Pi/(9*sqrt(3))) / sqrt(n)). - _Vaclav Kotesovec_, Aug 31 2015, extended Jan 21 2017
%F From _Michael Somos_, Oct 06 2019: (Start)
%F Expansion of q^(1/3) * eta(q^2) * eta(q^9) / (eta(q) * eta(q^18)) in powers of q.
%F Euler transform of period 18 sequence [1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...].
%F G.f. is a period 1 Fourier Series which satisifies f(-1 / (18 t)) = f(t) where q = exp(2 Pi i t).
%F Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (1 + u*v) * (u^3 + v^3) - u*v * (1 + u^2*v^2).
%F Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = (w^2 - v) * (u^2 - v) - 2*u*v*w.
%F Convolution inverse of A261733.
%F (End)
%e G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + ... _Michael Somos_, Oct 06 2019
%e G.f. = q^-1 + q^2 + q^5 + 2*q^8 + 2*q^11 + 3*q^14 + 4*q^17 + 5*q^20 + ...
%p b:= proc(n, i) option remember; local r;
%p `if`(2*n>i*(i+1)-(j-> 9*j*(j+1))(iquo(i, 9, 'r')), 0,
%p `if`(n=0, 1, b(n, i-1)+`if`(i>n or r=0, 0, b(n-i, i-1))))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..80); # _Alois P. Heinz_, Aug 31 2015
%t nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(9*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 31 2015 *)
%t b[n_, i_] := b[n, i] = Module[{q, r}, {q, r} = QuotientRemainder[i, 9]; If[2*n > i*(i+1) - 9*q*(q+1), 0, If[n == 0, 1, b[n, i-1] + If[i>n || r == 0, 0, b[n-i, i-1]]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Oct 07 2016, after _Alois P. Heinz_ *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^9] / (QPochhammer[ q] QPochhammer[ q^18]), {q, 0, n}]; (* _Michael Somos_, Oct 06 2019 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^9 + A) / (eta(x + A) * eta(x^18 + A)), n))}; /* _Michael Somos_, Oct 06 2019 */
%Y Cf. A261733.
%Y Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A261770 (m=6), A097793 (m=7), A261771 (m=8), A261772 (m=10).
%K nonn
%O 0,4
%A _Michael Somos_, Aug 28 2005
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