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)).

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, 126, 146, 168, 194, 222, 256, 291, 333, 379, 432, 489, 557, 629, 712, 805, 909, 1021, 1152, 1293, 1452, 1627, 1824, 2037, 2281, 2544, 2838, 3162, 3525, 3916, 4356

Offset: 0

Author

Paul D. Hanna, Jul 31 2022

Keywords

• nonn

Examples

			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 + ...
and the related series B(x) begins
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 + ...
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.

Crossrefs

Cf. A355359, A341714, A214157.

Programs

Mathematica

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 *)

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)};
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. A(x) = Sum_{n>=0} a(n)*x^n and related function B(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(Pi/13) * sin(3*Pi/13) * cos(5*Pi/26) * n^(3/4)). - Vaclav Kotesovec, Aug 01 2022