A359924 - OEIS

%I #21 Jan 05 2024 17:17:39

%S 1,2,26,372,6006,105338,1952102,37598422,745116966,15094772444,

%T 311183832004,6507065710068,137683172641240,2942394474649322,

%U 63418690179207242,1376986195691108990,30090726682472126472,661292884776232386766,14606177871231796042658,324062328994910188622258

%N a(n) = coefficient of x^n in A(x) such that 2/x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).

%H Paul D. Hanna, <a href="/A359924/b359924.txt">Table of n, a(n) for n = 1..200</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>.

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following.

%F (1) 2/x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).

%F (2) 2/x = Product_{n>=1} (1 - x^n) * (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^(2*n-1)*A(x)^2) * (1 - x^(2*n-1)/A(x)^2), by the Watson quintuple product identity.

%F a(n) = Sum_{k=0..n-1} A361050(n,k) * 2^k for n >= 1. - _Paul D. Hanna_, Mar 19 2023

%F a(n) ~ c * d^n / n^(3/2), where d = 24.0303544191480291910560326469... and c = 0.0066619562786442340995706184... - _Vaclav Kotesovec_, Mar 14 2023

%e G.f.: A(x) = x + 2*x^2 + 26*x^3 + 372*x^4 + 6006*x^5 + 105338*x^6 + 1952102*x^7 + 37598422*x^8 + 745116966*x^9 + 15094772444*x^10 + ...

%e where A = A(x) satisfies the doubly infinite sum

%e 2/x = ... + x^12*(1/A^9 - A^8) + x^5*(1/A^6 - A^5) + x*(1/A^3 - A^2) + (1 - 1/A) + x^2*(A^3 - 1/A^4) + x^7*(A^6 - 1/A^7) + x^15*(A^9 - 1/A^10) + ... + x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)) + ...

%e also, by the Watson quintuple product identity,

%e 2/x = (1-x)*(1-x*A)*(1-1/A)*(1-x*A^2)*(1-x/A^2) * (1-x^2)*(1-x^2*A)*(1-x/A)*(1-x^3*A^2)*(1-x^3/A^2) * (1-x^3)*(1-x^3*A)*(1-x^2/A)*(1-x^5*A^2)*(1-x^5/A^2) * (1-x^4)*(1-x^4*A)*(1-x^3/A)*(1-x^7*A^2)*(1-x^7/A^2) * ...

%o (PARI) /* Using the doubly infinite series */

%o {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);

%o A[#A] = polcoeff(2/x - sum(m=-#A,#A, (Ser(A)^(3*m) - 1/Ser(A)^(3*m+1)) * x^(m*(3*m+1)/2) ),#A-4) ); A[n+1]}

%o for(n=1,30, print1(a(n),", "))

%o (PARI) /* Using the quintuple product */

%o {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);

%o A[#A] = polcoeff(2/x - prod(m=1,#A, (1 - x^m) * (1 - x^m*Ser(A)) * (1 - x^(m-1)/Ser(A)) * (1 - x^(2*m-1)*Ser(A)^2) * (1 - x^(2*m-1)/Ser(A)^2) ),#A-4) ); A[n+1]}

%o for(n=1,30, print1(a(n),", "))

%Y Cf. A359920, A359921, A361050, A361052, A361538.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 22 2023