A355359 - OEIS

%I #12 Aug 02 2022 10:38:59

%S 1,0,1,0,1,1,2,2,3,2,4,4,6,6,8,9,11,12,16,17,22,24,29,32,39,43,53,57,

%T 69,75,90,99,117,129,150,166,193,213,246,273,312,346,394,436,496,549,

%U 621,687,774,855,962,1062,1192,1313,1470,1618,1807,1989,2214,2436

%N Coefficients in the expansion of 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)).

%H Seiichi Manyama, <a href="/A355359/b355359.txt">Table of n, a(n) for n = 0..10000</a>

%H 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 G.f. B(x) = Sum_{n>=0} a(n)*x^n and related function A(x) satisfy the following relations.

%F (1.a) A(x^3)*B(x) - x^2*A(x)*B(x^3) = 1.

%F (1.b) A(x)^3*B(x) - x^2*A(x)*B(x)^3 - 3*x*A(x)^2*B(x)^2 = 1.

%F (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 (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 (3.a) A(x)*B(x) = Product_{n>=1} (1 - x^(13*n))/(1 - x^n), a g.f. of A341714.

%F (3.b) A(x)/B(x) = Product_{n>=1} 1/(1 - x^n)^Kronecker(13, n), a g.f. of A214157.

%F (4.a) A(x) = sqrt( Product_{n>=1} (1 - x^(13*n))/(1 - x^n)^(1 + Kronecker(13, n)) ).

%F (4.b) B(x) = sqrt( Product_{n>=1} (1 - x^(13*n))/(1 - x^n)^(1 - Kronecker(13, n)) ).

%F (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 (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 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 a(n) ~ exp(2*Pi*sqrt(n/13)) / (16 * 13^(1/4) * sin(2*Pi/13) * cos(Pi/26) * cos(3*Pi/26) * n^(3/4)). - _Vaclav Kotesovec_, Aug 01 2022

%e G.f.: 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 + ...

%e and the related series A(x) begins

%e 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 + ... + A355358(n)*x^n + ...

%e such that A(x) and B(x) satisfy

%e 1 = A(x^3)*B(x) - x^2*A(x)*B(x^3),

%e and

%e 1 = A(x)^3*B(x) - x^2*A(x)*B(x)^3 - 3*x*A(x)^2*B(x)^2.

%e Related expansions begin

%e 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 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 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 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 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 nmax = 60; CoefficientList[Series[1/Product[(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)), {n, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 01 2022 *)

%o (PARI) {a(n) = polcoeff( 1/prod(m=0, n, (1 - x^(13*m+2))*(1 - x^(13*m+5))*(1 - x^(13*m+6))*(1 - x^(13*m+7))*(1 - x^(13*m+8))*(1 - x^(13*m+11)), 1 + x*O(x^n)), n)};

%o for(n=0,60,print1(a(n),", "))

%o (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 for(n=0,60,print1(a(n),", "))

%Y Cf. A355358, A341714, A214157.

%K nonn

%O 0,7

%A _Paul D. Hanna_, Aug 01 2022