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%I #17 Nov 13 2016 05:24:04
%S 1,1,3,6,11,18,31,49,78,119,180,267,394,567,813,1151,1616,2244,3099,
%T 4240,5769,7790,10462,13965,18552,24502,32223,42176,54972,71340,92242,
%U 118800,152481,195017,248621,315945,400315,505694,637068,800380,1002964
%N The number phi_3(n) of Frobenius partitions that allow up to 3 repetitions of an integer in a row.
%D Andrews, George E., Generalized Frobenius partitions, Memoirs of the American Mathematical Society, Number 301, May 1984. See phi_3(n), page 41.
%F Expansion of q^(1/8) * eta(q^6)^5 / (eta(q) * eta(q^2) * eta(q^3)^2 * eta(q^12)^2) in powers of q. - _Michael Somos_, Mar 09 2011
%F Euler transform of period 12 sequence [ 1, 2, 3, 2, 1, -1, 1, 2, 3, 2, 1, 1, ...]. - _Michael Somos_, Mar 09 2011
%F G.f.: Product_{k>0} (1 - x^(12*k - 6)) / ( (1 - x^(6*k - 5)) * (1 - x^(6*k - 4))^2 * (1 - x^(6*k - 3))^3 * (1 - x^(6*k - 2))^2 * (1 - x^(6*k - 1)) *(1 - x^(12*k)) ). [Andrews, p. 11, equation (5.10)]
%F a(n) ~ exp(sqrt(n)*Pi)/(4*sqrt(6)*n). - _Vaclav Kotesovec_, Nov 13 2016
%e 1 + x + 3*x^2 + 6*x^3 + 11*x^4 + 18*x^5 + 31*x^6 + 49*x^7 + 78*x^8 + ...
%e 1/q + q^7 + 3*q^15 + 6*q^23 + 11*q^31 + 18*q^39 + 31*q^47 + 49*q^55 + 78*q^63 + ...
%t nmax = 50; CoefficientList[Series[Product[(1 + x^(6*k - 3)) / ( (1 - x^(6*k - 5)) * (1 - x^(6*k - 4))^2 * (1 - x^(6*k - 3))^2 * (1 - x^(6*k - 2))^2 * (1 - x^(6*k - 1)) * (1 - x^(12*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 13 2016 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^5 / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2), n))} /* _Michael Somos_, Mar 09 2011 */
%K easy,nonn
%O 0,3
%A _James A. Sellers_, Apr 04 2000