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A337949 a(n) = 2^(n*(n-1)/2) + 2^(n*(n+1)/2) for n > 0, with a(0) = 1. 5
1, 3, 10, 72, 1088, 33792, 2129920, 270532608, 68987912192, 35253091565568, 36063981391052800, 73823005091857170432, 302305241879952131883008, 2476182310025664207091924992, 40567295087381911608444300820480, 1329268560604123176244654954782916608 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..15.

FORMULA

G.f.: Sum_{n=-oo..+oo} 2^(n*(n+1)/2) * x^(n^2) = Sum_{n>=0} a(n) * x^(n^2).

G.f.: Product_{n>=1} (1 - 2^n*x^(2*n)) * (1 + 2^n*x^(2*n-1)) * (1 + 2^(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 3), a(4*n+2) = 1 (mod 3), a(4*n+4) = 2 (mod 3), for n >= 0 (conjecture).

EXAMPLE

G.f.: A(x) = 1 + 3*x + 10*x^4 + 72*x^9 + 1088*x^16 + 33792*x^25 + 2129920*x^36 + 270532608*x^49 + 68987912192*x^64 + 35253091565568*x^81 + 36063981391052800*x^100 + ... + a(n)*x^(n^2) + ...

which can be generated by the Jacobi Triple Product:

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

PROG

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

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

(PARI) /* As Coefficients in a Jacobi Theta Function: */

{a(n) = polcoeff( sum(m=-n, n, 2^(m*(m+1)/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\2+1, (1 - 2^m*x^(2*m)) * (1 + 2^m*x^(2*m-1)) * (1 + 2^(m-1)*x^(2*m-1)) +x*O(x^(n^2))), n^2)}

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

CROSSREFS

Cf. A337948, A337951.

Sequence in context: A181075 A136507 A334211 * A086846 A277614 A082245

Adjacent sequences:  A337946 A337947 A337948 * A337950 A337951 A337952

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 03 2020

STATUS

approved

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Last modified June 23 08:51 EDT 2021. Contains 345395 sequences. (Running on oeis4.)