A355359 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)).
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, 69, 75, 90, 99, 117, 129, 150, 166, 193, 213, 246, 273, 312, 346, 394, 436, 496, 549, 621, 687, 774, 855, 962, 1062, 1192, 1313, 1470, 1618, 1807, 1989, 2214, 2436
Offset: 0
Keywords
Examples
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 + ... and the related series A(x) begins 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 + ... such that A(x) and B(x) satisfy 1 = A(x^3)*B(x) - x^2*A(x)*B(x^3), and 1 = A(x)^3*B(x) - x^2*A(x)*B(x)^3 - 3*x*A(x)^2*B(x)^2. Related expansions begin 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 + ... 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 + ... 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 + ... 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 + ... 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 + ...
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000
- Srinivasa Ramanujan, Algebraic relations between certain infinite products, Proceedings of the London Mathematical Society, vol.2, no.18, 1920.
Programs
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Mathematica
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 *) -
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)}; for(n=0,60,print1(a(n),", ")) -
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)}; for(n=0,60,print1(a(n),", "))
Formula
G.f. B(x) = Sum_{n>=0} a(n)*x^n and related function A(x) satisfy the following relations.
(1.a) A(x^3)*B(x) - x^2*A(x)*B(x^3) = 1.
(1.b) A(x)^3*B(x) - x^2*A(x)*B(x)^3 - 3*x*A(x)^2*B(x)^2 = 1.
(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)).
(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)).
(3.a) A(x)*B(x) = Product_{n>=1} (1 - x^(13*n))/(1 - x^n), a g.f. of A341714.
(3.b) A(x)/B(x) = Product_{n>=1} 1/(1 - x^n)^Kronecker(13, n), a g.f. of A214157.
(4.a) A(x) = sqrt( Product_{n>=1} (1 - x^(13*n))/(1 - x^n)^(1 + Kronecker(13, n)) ).
(4.b) B(x) = sqrt( Product_{n>=1} (1 - x^(13*n))/(1 - x^n)^(1 - Kronecker(13, n)) ).
(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) ).
(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) ) .
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..
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