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%I A213598 #33 Jun 16 2017 07:54:31
%S A213598 1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627,792,
%T A213598 1002,1255,1575,1958,2436,3010,3718,4565,5604,6842,8349,10143,12310,
%U A213598 14883,17977,21637,26015,31185,37338,44583,53174,63261,75175,89134,105558,124754,147273,173524
%N A213598 Number of partitions of n in which no parts are multiples of 49.
%C A213598 For n<49 we have a(n)=A000041(n), for n>=49 a(n)!=A000041(n).
%C A213598 In Fricke page 401, he gives the expansion sigma(omega) = q^4 + q^6 + 2q^8 + 3q^10 + 5q^12 + 7q^14 + 11q^16 + 15q^18 + ... where q = exp( Pi i omega).
%D A213598 R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 401. Eq. (49)
%H A213598 Seiichi Manyama, <a href="/A213598/b213598.txt">Table of n, a(n) for n = 0..10000</a>
%H A213598 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015.
%F A213598 Expansion of q^(-2) * eta(q^49) / eta(q) in powers of q.
%F A213598 Euler transform of period 49 sequence [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, ...].
%F A213598 Given g.f. A(x) then B(x) = x^2 * A(x) satisfies 0 = f(B(x), B(x^2),
%F A213598 B(x^4)) where f(u, v, w) = u * v * w * (1 - 7*v^2) - (v - w) * (u - v) * (v^2 - u*w).
%F A213598 G.f. is a period 1 Fourier series which satisfies f(-1 / (49 t)) = 1 / (7 f(t)) where q = exp(2 Pi i t).
%F A213598 G.f.: Product_{k>0} (1 - x^(49*k)) / (1 - x^k).
%F A213598 a(n) ~ exp(4*Pi*sqrt(2*n)/7) / (2^(1/4) * 7^(3/2) * n^(3/4)). - _Vaclav Kotesovec_, Oct 14 2015
%F A213598 a(n) = (1/n)*Sum_{k=1..n} A287926(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Jun 16 2017
%e A213598 G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + ...
%e A213598 G.f. = q^2 + q^3 + 2*q^4 + 3*q^5 + 5*q^6 + 7*q^7 + 11*q^8 + 15*q^9 + 22*q^10 + ...
%t A213598 a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 49, n, 49}] / Product[ 1 - x^k, {k, n}], {x, 0, n}];
%t A213598 a[ n_] := SeriesCoefficient[ QPochhammer[ x^49] / QPochhammer[ x], {x, 0, n}]; (* _Michael Somos_, May 13 2014 *)
%o A213598 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^49 + A) / eta(x + A), n))};
%Y A213598 Cf. A000009 (m=2), A000726 (m=3), A001935 (m=4), A035959 (m=5), A219601 (m=6), A035985 (m=7), A261775 (m=8), A104502 (m=9), A261776 (m=10), A092885 (m=25), this sequence (m=49).
%Y A213598 Cf. A000041, A287926.
%K A213598 nonn
%O A213598 0,3
%A A213598 _Michael Somos_, Jun 14 2012

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