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A337951
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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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5
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1, 5, 68, 4160, 1052672, 1074790400, 4399120252928, 72061992084439040, 4722438540463683141632, 1237944761651863144544337920, 1298075452573746192512898981429248, 5444519168809230049120900851532373688320, 91343857777699303122745717458761407636787167232
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f.: Sum_{n=-oo..+oo} 2^n * (2*x)^(n^2) = Sum_{n>=0} a(n) * x^(n^2).
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.
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).
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EXAMPLE
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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) + ...
which can be generated by the Jacobi Triple Product:
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)) * ...
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MATHEMATICA
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Join[{1}, Total[4^#]&/@Partition[Accumulate[Range[0, 15]], 2, 1]] (* Harvey P. Dale, Oct 12 2021 *)
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PROG
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(PARI) {a(n) = if(n==0, 1, 2^(n*(n-1)) + 2^(n*(n+1)))}
for(n=0, 15, print1(a(n), ", "))
(PARI) /* As Coefficients in a Jacobi Theta Function: */
{a(n) = polcoeff( sum(m=-n, n, 2^m*(2*x)^(m^2) +x*O(x^(n^2))), n^2)}
for(n=0, 15, print1(a(n), ", "))
(PARI) /* By the Jacobi Triple Product identity: */
{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)}
for(n=0, 15, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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