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A355358 Coefficients in the expansion of A(x) = 1 / Product_{n>=0} (1 - x^(13*n+1))*(1 - x^(13*n+3))*(1 - x^(13*n+4))*(1 - x^(13*n+9))*(1 - x^(13*n+10))*(1 - x^(13*n+12)).

%I A355358 #12 Aug 02 2022 10:38:43
%S A355358 1,1,1,2,3,3,4,5,6,8,10,11,15,18,21,25,31,36,43,50,59,69,81,93,109,
%T A355358 126,146,168,194,222,256,291,333,379,432,489,557,629,712,805,909,1021,
%U A355358 1152,1293,1452,1627,1824,2037,2281,2544,2838,3162,3525,3916,4356
%N A355358 Coefficients in the expansion of A(x) = 1 / Product_{n>=0} (1 - x^(13*n+1))*(1 - x^(13*n+3))*(1 - x^(13*n+4))*(1 - x^(13*n+9))*(1 - x^(13*n+10))*(1 - x^(13*n+12)).
%H A355358 Seiichi Manyama, <a href="/A355358/b355358.txt">Table of n, a(n) for n = 0..10000</a>
%H A355358 Srinivasa Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper29/page1.htm">Algebraic relations between certain infinite products</a>, Proceedings of the London Mathematical Society, vol.2, no.18, 1920.
%F A355358 G.f. A(x) = Sum_{n>=0} a(n)*x^n and related function B(x) satisfy the following relations.
%F A355358 (1.a) A(x^3)*B(x) - x^2*A(x)*B(x^3) = 1.
%F A355358 (1.b) A(x)^3*B(x) - x^2*A(x)*B(x)^3 - 3*x*A(x)^2*B(x)^2 = 1.
%F A355358 (2.a) A(x) = 1 / Product_{n>=0} (1 - x^(13*n+1))*(1 - x^(13*n+3))*(1 - x^(13*n+4))*(1 - x^(13*n+9))*(1 - x^(13*n+10))*(1 - x^(13*n+12)).
%F A355358 (2.b) B(x) = 1 / Product_{n>=0} (1 - x^(13*n+2))*(1 - x^(13*n+5))*(1 - x^(13*n+6))*(1 - x^(13*n+7))*(1 - x^(13*n+8))*(1 - x^(13*n+11)).
%F A355358 (3.a) A(x)*B(x) = Product_{n>=1} (1 - x^(13*n))/(1 - x^n), a g.f. of A341714.
%F A355358 (3.b) A(x)/B(x) = Product_{n>=1} 1/(1 - x^n)^Kronecker(13, n), a g.f. of A214157.
%F A355358 (4.a) A(x) = sqrt( Product_{n>=1} (1 - x^(13*n))/(1 - x^n)^(1 + Kronecker(13, n)) ).
%F A355358 (4.b) B(x) = sqrt( Product_{n>=1} (1 - x^(13*n))/(1 - x^n)^(1 - Kronecker(13, n)) ).
%F A355358 (5.a) A(x) = ( Product_{n>=1} (1 - x^(13*n))^3 ) / ( f(-x, -x^12) * f(-x^3, -x^10) * f(-x^4, -x^9) ).
%F A355358 (5.b) B(x) = ( Product_{n>=1} (1 - x^(13*n))^3 ) / ( f(-x^2, -x^11) * f(-x^5, -x^8) * f(-x^6, -x^7) ) .
%F A355358 Formulas (4.*) and (5.*) are derived from formulas given by _Michael Somos_ in A214157, where f(a,b) = Sum_{n=-oo..+oo} a^(n*(n+1)/2) * b^(n*(n-1)/2) is Ramanujan's theta function..
%F A355358 a(n) ~ exp(2*Pi*sqrt(n/13)) / (16 * 13^(1/4) * sin(Pi/13) * sin(3*Pi/13) * cos(5*Pi/26) * n^(3/4)). - _Vaclav Kotesovec_, Aug 01 2022
%e A355358 G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + 8*x^9 + 10*x^10 + 11*x^11 + 15*x^12 + 18*x^13 + 21*x^14 + ...
%e A355358 and the related series B(x) begins
%e A355358 B(x) = 1 + x^2 + x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 2*x^9 + 4*x^10 + 4*x^11 + 6*x^12 + 6*x^13 + 8*x^14 + 9*x^15 + ... + A355359(n)*x^n + ...
%e A355358 such that A(x) and B(x) satisfy
%e A355358 1 = A(x^3)*B(x) - x^2*A(x)*B(x^3),
%e A355358 and
%e A355358 1 = A(x)^3*B(x) - x^2*A(x)*B(x)^3 - 3*x*A(x)^2*B(x)^2.
%e A355358 Related expansions begin
%e A355358 A(x^3)*B(x) = 1 + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 6*x^9 + 7*x^10 + 10*x^11 + 13*x^12 + 14*x^13 + 20*x^14 + 24*x^15 + ...
%e A355358 A(x)*B(x^3) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 5*x^6 + 6*x^7 + 7*x^8 + 10*x^9 + 13*x^10 + ...
%e A355358 A(x)^3*B(x) = 1 + 3*x + 7*x^2 + 16*x^3 + 34*x^4 + 65*x^5 + 120*x^6 + 213*x^7 + 365*x^8 + 609*x^9 + 994*x^10 + ...
%e A355358 A(x)*B(x)^3 = 1 + x + 4*x^2 + 5*x^3 + 12*x^4 + 18*x^5 + 35*x^6 + 54*x^7 + 94*x^8 + 142*x^9 + 232*x^10 + ...
%e A355358 A(x)^2*B(x)^2 = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 65*x^6 + 110*x^7 + 185*x^8 + 300*x^9 + 481*x^10 + ...
%t A355358 nmax = 60; CoefficientList[Series[1/Product[(1 - x^(13*n + 1))*(1 - x^(13*n + 3))*(1 - x^(13*n + 4))*(1 - x^(13*n + 9))*(1 - x^(13*n + 10))*(1 - x^(13*n + 12)), {n, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 01 2022 *)
%o A355358 (PARI) {a(n) = polcoeff( 1/prod(m=0, n, (1 - x^(13*m+1))*(1 - x^(13*m+3))*(1 - x^(13*m+4))*(1 - x^(13*m+9))*(1 - x^(13*m+10))*(1 - x^(13*m+12)), 1 + x*O(x^n)), n)};
%o A355358 for(n=0,60,print1(a(n),", "))
%o A355358 (PARI) {a(n) = polcoeff( sqrt( prod(k=1, n, (1 - x^(13*k))/(1 - x^k)^(1 + kronecker(13, k)), 1 + x*O(x^n)) ), n)};
%o A355358 for(n=0,60,print1(a(n),", "))
%Y A355358 Cf. A355359, A341714, A214157.
%K A355358 nonn
%O A355358 0,4
%A A355358 _Paul D. Hanna_, Jul 31 2022