|
%I #14 Jan 05 2024 17:17:22
%S 1,-11,58,-225,2146,-14821,85590,-1974433,9180658,2927259,-85838114,
%T 63964584095,-520091681238,16934937109019,-384678052715594,
%U 5238404820228159,-295855770548974622,4600244140822151099,-186350295911412573810,4851711966859680480959
%N a(n) = coefficient of x^n/n! in A(x) = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (exp(3*n*x) - exp(-(3*n+1)*x)).
%H Paul D. Hanna, <a href="/A359719/b359719.txt">Table of n, a(n) for n = 1..400</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>.
%F E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! may be defined by the following.
%F (1) A(x) = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (exp(3*n*x) - exp(-(3*n+1)*x)).
%F (2) A(x) = Product_{n>=1} (1 - x^n) * (1 - x^n*exp(x)) * (1 - x^(n-1)*exp(-x)) * (1 - x^(2*n-1)*exp(2*x)) * (1 - x^(2*n-1)*exp(-2*x)), by the Watson quintuple product identity.
%F (3) A(x) = 2*exp(-x/2) * Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * sinh((6*n+1)*x/2).
%F (4) A(x) = (1 - exp(-x)) * Product_{n>=1} (1 - x^n) * (1 - 2*x^n*cosh(x) + x^(2*n)) * (1 - 2*x^(2*n-1)*cosh(2*x) + x^(4*n-2)).
%e E.g.f.: A(x) = x - 11*x^2/2! + 58*x^3/3! - 225*x^4/4! + 2146*x^5/5! - 14821*x^6/6! + 85590*x^7/7! - 1974433*x^8/8! + 9180658*x^9/9! + 2927259*x^10/10! + ...
%e where A(x) equals the doubly infinite series
%e A(x) = ... + x^12*(exp(-9*x) - exp(8*x)) + x^5*(exp(-6*x) - exp(5*x)) + x*(exp(-3*x) - exp(2*x)) + (1 - exp(-x)) + x^2*(exp(3*x) - exp(-4*x)) + x^7*(exp(6*x) - exp(-7*x)) + x^15*(exp(9*x) - exp(-10*x)) + ... + x^(n*(3*n+1)/2) * (exp(3*n*x) - exp(-(3*n+1)*x)) + ...
%e also, by the Watson quintuple product identity,
%e A(x) = (1-x)*(1-x*exp(x))*(1-1*exp(-x))*(1-x*exp(2*x))*(1-x*exp(-2*x)) * (1-x^2)*(1-x^2*exp(x))*(1-x*exp(-x))*(1-x^3*exp(2*x))*(1-x^3*exp(-2*x)) * (1-x^3)*(1-x^3*exp(x))*(1-x^2*exp(-x))*(1-x^5*exp(2*x))*(1-x^5*exp(-2*x)) * (1-x^4)*(1-x^4*exp(x))*(1-x^3*exp(-x))*(1-x^7*exp(2*x))*(1-x^7*exp(-2*x)) * ...
%o (PARI) /* Using the doubly infinite series */
%o {a(n) = my(X=x+x*O(x^n),M=sqrtint(2*n)); n! * polcoeff( sum(m=-M,M, x^(m*(3*m+1)/2) * (exp(3*m*X) - exp(-(3*m+1)*X)) ), n)}
%o for(n=1,30, print1(a(n),", "))
%o (PARI) /* Using the quintuple product */
%o {a(n) = my(X=x+x*O(x^n)); n! * polcoeff( prod(m=1,n, (1 - x^m) * (1 - x^m*exp(X)) * (1 - x^(m-1)*exp(-X)) * (1 - x^(2*m-1)*exp(2*X)) * (1 - x^(2*m-1)*exp(-2*X)) ),n)}
%o for(n=1,30, print1(a(n),", "))
%Y Cf. A359920, A359919.
%K sign
%O 1,2
%A _Paul D. Hanna_, Jan 22 2023
| 