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A282937 a(n) = A000728(5*n).

%I A282937 #50 Feb 16 2025 08:33:42
%S A282937 1,-6,9,10,-30,1,5,51,10,-100,20,-55,109,110,-130,-1,-110,160,10,-230,
%T A282937 100,15,191,120,-230,-100,-89,160,90,-340,120,5,300,200,-260,-1,-275,
%U A282937 240,-100,-270,119,-165,260,410,-200,-40,20,200,-110,-500,180,-54,140
%N A282937 a(n) = A000728(5*n).
%C A282937 This generalized function is related to two following identities; R(q^5) - q - q^2/R(q^5) = (q; q)_{infinity}/(q^25; q^25)_{infinity}, R^5(q^5) - 11*q^5 - q^10/R^5(q^5) = ((q^5; q^5)_{infinity}/(q^25; q^25)_{infinity})^6, where R(q) is the Rogers-Ramanujan continued function and (q; q)_n is the q-Pochhammer symbol. See the reference.
%D A282937 G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 185.
%H A282937 Seiichi Manyama, <a href="/A282937/b282937.txt">Table of n, a(n) for n = 0..1000</a>
%H A282937 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>
%F A282937 G.f.: Product_{n>=1} (1 - q^n)^6/(1 - q^(5*n)).
%F A282937 a(n) = (-1)^j mod 5 if n = j*(3*j - 1)/2 for all j in Z; otherwise a(n) = 0 mod 5.
%F A282937 Sum_{k=0..n} a(k)*A182821(n-k) = 0 for n > 0. - _Seiichi Manyama_, Feb 28 2017
%F A282937 G.f.: exp( Sum_{n>=1} -sigma(5*n)*q^n/n ). - _Seiichi Manyama_, Mar 04 2017
%F A282937 a(n) = -(1/n)*Sum_{k=1..n} sigma(5*k)*a(n-k). - _Seiichi Manyama_, Mar 04 2017
%e A282937 G.f.: 1 - 6*q + 9*q^2 + 10*q^3 - 30*q^4 + q^5 + 5*q^6 + 51*q^7 + ...
%t A282937 CoefficientList[Series[Product[(1 - x^j)^6/(1 - x^(5*j)), {j,1,62}], {x,0,60}], x] (* _G. C. Greubel_, Nov 18 2018 *)
%o A282937 (PARI) m=60; x='x+O('x^m); Vec(prod(j=1,m+2, (1 - x^j)^6/(1 - x^(5*j)))) \\ _G. C. Greubel_, Nov 18 2018
%o A282937 (Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^j)^6/(1 - x^(5*j)): j in [1..(m+2)]]) )); // _G. C. Greubel_, Nov 18 2018
%o A282937 (Sage)
%o A282937 R = PowerSeriesRing(ZZ, 'x')
%o A282937 prec = 60
%o A282937 x = R.gen().O(prec)
%o A282937 s = prod((1 - x^j)^6/(1 - x^(5*j)) for j in (1..prec))
%o A282937 print(s.coefficients()) # _G. C. Greubel_, Nov 18 2018
%Y A282937 Cf. A000728, A182821 (Product_{n>=1} (1 - q^(5*n))/(1 - q^n)^6), A282941.
%Y A282937 Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), A115110 (k=2), A185654 (k=3), this sequence (k=5), A282942 (k=7).
%K A282937 sign
%O A282937 0,2
%A A282937 _Seiichi Manyama_, Feb 25 2017