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a(n) = 4^(n*(n-1)/2) + 4^(n*(n+1)/2) for n > 0, with a(0) = 1.
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%I #9 Oct 12 2021 17:03:18

%S 1,5,68,4160,1052672,1074790400,4399120252928,72061992084439040,

%T 4722438540463683141632,1237944761651863144544337920,

%U 1298075452573746192512898981429248,5444519168809230049120900851532373688320,91343857777699303122745717458761407636787167232

%N a(n) = 4^(n*(n-1)/2) + 4^(n*(n+1)/2) for n > 0, with a(0) = 1.

%H Harvey P. Dale, <a href="/A337951/b337951.txt">Table of n, a(n) for n = 0..57</a>

%F G.f.: Sum_{n=-oo..+oo} 2^n * (2*x)^(n^2) = Sum_{n>=0} a(n) * x^(n^2).

%F G.f.: Product_{n>=1} (1 - 4^n*x^(2*n)) * (1 + 4^n*x^(2*n-1)) * (1 + 4^(n-1)*x^(2*n-1)) = Sum_{n>=0} a(n) * x^(n^2), by the Jacobi triple product identity.

%F a(2*n+1) = 0 (mod 5), a(4*n+2) = 3 (mod 5), a(4*n+4) = 2 (mod 5), for n >= 0 (conjecture).

%e G.f.: A(x) = 1 + 5*x + 68*x^4 + 4160*x^9 + 1052672*x^16 + 1074790400*x^25 + 4399120252928*x^36 + 72061992084439040*x^49 + 4722438540463683141632*x^64 + 1237944761651863144544337920*x^81 + ... + a(n)*x^(n^2) + ...

%e which can be generated by the Jacobi Triple Product:

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

%t Join[{1},Total[4^#]&/@Partition[Accumulate[Range[0,15]],2,1]] (* _Harvey P. Dale_, Oct 12 2021 *)

%o (PARI) {a(n) = if(n==0,1, 2^(n*(n-1)) + 2^(n*(n+1)))}

%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, 2^m*(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, (1 - 4^m*x^(2*m)) * (1 + 4^m*x^(2*m-1)) * (1 + 4^(m-1)*x^(2*m-1)) +x*O(x^(n^2))),n^2)}

%o for(n=0,15,print1(a(n),", "))

%Y Cf. A337950, A337949.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 03 2020