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%I #4 Oct 04 2020 20:23:21
%S 1,6,130,15750,9781250,30527343750,476867675781250,
%T 37253379821777343750,14551952481269836425781250,
%U 28421723982319235801696777343750,277555784577998565509915351867675781250,13552527433624561581382295116782188415527343750
%N a(n) = 5^(n*(n-1)/2) + 5^(n*(n+1)/2) for n > 0, with a(0) = 1.
%F G.f.: Sum_{n=-oo..+oo} 5^(n*(n+1)/2) * x^(n^2) = Sum_{n>=0} a(n) * x^(n^2).
%F G.f.: Product_{n>=1} (1 - 5^n*x^(2*n)) * (1 + 5^n*x^(2*n-1)) * (1 + 5^(n-1)*x^(2*n-1)) = Sum_{n>=0} a(n) * x^(n^2), by the Jacobi triple product identity.
%e G.f.: A(x) = 1 + 6*x + 130*x^4 + 15750*x^9 + 9781250*x^16 + 30527343750*x^25 + 476867675781250*x^36 + 37253379821777343750*x^49 + 14551952481269836425781250*x^64 + ... + a(n)*x^(n^2) + ...
%e which can be generated by the Jacobi Triple Product:
%e A(x) = (1 - 5*x^2)*(1 + 5*x)*(1 + x) * (1 - 5^2*x^4)*(1 + 5^2*x^3)*(1 + 5*x^3) * (1 - 5^3*x^6)*(1 + 5^3*x^5)*(1 + 5^2*x^5) * (1 - 5^4*x^8)*(1 + 5^4*x^7)*(1 + 5^3*x^7) * ... * (1 - 5^n*x^(2*n))*(1 + 5^n*x^(2*n-1))*(1 + 5^(n-1)*x^(2*n-1)) * ...
%o (PARI) {a(n) = if(n==0,1, 5^(n*(n-1)/2) + 5^(n*(n+1)/2))}
%o for(n=0,15,print1(a(n),", "))
%o (PARI) /* As Coefficients in a Jacobi Theta Function: */
%o {a(n) = polcoeff( sum(m=-n,n, 5^(m*(m+1)/2)*x^(m^2) +x*O(x^(n^2))),n^2)}
%o for(n=0,15,print1(a(n),", "))
%o (PARI) /* By the Jacobi Triple Product identity: */
%o {a(n) = polcoeff( prod(m=1,n^2\2+1, (1 - 5^m*x^(2*m)) * (1 + 5^m*x^(2*m-1)) * (1 + 5^(m-1)*x^(2*m-1)) +x*O(x^(n^2))),n^2)}
%o for(n=0,15,print1(a(n),", "))
%Y Cf. A337970, A337949, A337951, A337969.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Oct 04 2020
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