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 A112193 Coefficients of replicable function number "54b". 9
 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, 127, 145, 167, 191, 218, 249, 283, 322, 365, 414, 469, 529, 597, 673, 757, 851, 955, 1071, 1199, 1341, 1499, 1673, 1865, 2078, 2313, 2572, 2857, 3171, 3517, 3897 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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 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 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..10000 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra 22, No. 13, 5175-5193 (1994). FORMULA 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 From Michael Somos, Oct 06 2019: (Start) Expansion of q^(1/3) * eta(q^2) * eta(q^9) / (eta(q) * eta(q^18)) in powers of q. Euler transform of period 18 sequence [1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...]. G.f. is a period 1 Fourier Series which satisifies f(-1 / (18 t)) = f(t) where q = exp(2 Pi i t). 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). 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. Convolution inverse of A261733. (End) EXAMPLE 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 G.f. = q^-1 + q^2 + q^5 + 2*q^8 + 2*q^11 + 3*q^14 + 4*q^17 + 5*q^20 + ... MAPLE b:= proc(n, i) option remember;  local r;       `if`(2*n>i*(i+1)-(j-> 9*j*(j+1))(iquo(i, 9, 'r')), 0,       `if`(n=0, 1, b(n, i-1)+`if`(i>n or r=0, 0, b(n-i, i-1))))     end: a:= n-> b(n\$2): seq(a(n), n=0..80);  # Alois P. Heinz, Aug 31 2015 MATHEMATICA nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(9*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *) 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 *) a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^9] / (QPochhammer[ q] QPochhammer[ q^18]), {q, 0, n}]; (* Michael Somos, Oct 06 2019 *) PROG (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 */ CROSSREFS Cf. A261733. 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). Sequence in context: A304631 A298603 A134345 * A014215 A067014 A096791 Adjacent sequences:  A112190 A112191 A112192 * A112194 A112195 A112196 KEYWORD nonn,changed AUTHOR Michael Somos, Aug 28 2005 STATUS approved

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Last modified October 16 13:51 EDT 2019. Contains 328093 sequences. (Running on oeis4.)