A256357 L.g.f.: log( 1 + Sum_{n>=1} x^(n^2) + x^(2*n^2) ).
1, 1, -2, 5, -4, -2, 8, -3, 7, -4, -10, 14, -12, 8, 8, -19, 18, 7, -18, 20, -16, -10, 24, -18, 21, -12, -20, 40, -28, 8, 32, -51, 20, 18, -32, 59, -36, -18, 24, -28, 42, -16, -42, 38, -28, 24, 48, -82, 57, 21, -36, 44, -52, -20, 40, -24, 36, -28, -58, 104, -60, 32, 56, -115, 48, 20, -66, 90, -48, -32, 72, -45, 74, -36, -42, 62, -80, 24, 80, -124, 61, 42, -82, 112, -72, -42, 56, -58, 90, -28, -96, 120, -64, 48, 72, -210, 98, 57, -70, 145, -100, -36, 104, -68, 64, -52, -106, 140, -108, 40, 72, -152, 114, 36, -96, 92, -84, -58, 144, -88, 111
Offset: 1
Keywords
Examples
L.g.f.: L(x) = x + x^2/2 - 2*x^3/3 + 5*x^4/4 - 4*x^5/5 - 2*x^6/6 + 8*x^7/7 - 3*x^8/8 + 7*x^9/9 - 4*x^10/10 - 10*x^11/11 + 14*x^12/12 - 12*x^13/13 + 8*x^14/14 + 8*x^15/15 - 19*x^16/16 +...+ a(n)*x^n/n +... where exp(L(x)) = 1 + x + x^2 + x^4 + x^8 + x^9 + x^16 + x^18 + x^25 + x^32 + x^36 + x^49 + x^50 + x^64 + x^72 + x^81 + x^98 + x^100 +...+ x^A028982(n) +...
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..1024
- Cooper, Shaun; Hirschhorn, Michael. On Some Finite Product Identities. Rocky Mountain J. Math. 31 (2001), no. 1, 131--139.
Programs
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PARI
{a(n) = local(L=x); L = log(1 + sum(k=1,sqrtint(n+1), x^(k^2) + x^(2*k^2)) +x*O(x^n)); n*polcoeff(L,n)} for(n=1,121, print1(a(n),", ")) -
PARI
{a(n) = -sigma(n) + sumdiv(n,d, if(d%4==2,d)) + 2*sumdiv(n,d, if((d%8)%3==1,d))} for(n=1,121, print1(a(n),", ")) -
PARI
{a(n) = local(L, X=x+x*O(x^n)); L = sum(m=0,n\8+1, log( (1-x^(3+8*m))*(1-x^(5+8*m))*(1-x^(8+8*m)) / ( (1-x^(1+8*m))*(1-x^(4+8*m))*(1-x^(7+8*m) +x*O(x^n)) ))); n*polcoeff(L,n)} for(n=1,121, print1(a(n),", "))
Formula
L.g.f.: Sum_{n>=0} log( (1-x^(3+8*n))*(1-x^(5+8*n))*(1-x^(8+8*n)) / ( (1-x^(1+8*n))*(1-x^(4+8*n))*(1-x^(7+8*n)) ) ). [See Cooper and Hirschhorn reference]
From formulas given by Michael Somos in A093709: (Start)
L.g.f.: log( (theta_3(x) + theta_3(x^2)) / 2).
L.g.f.: Log( psi(q^4) * f(-q^3, -q^5) / f(-q, -q^7) ) in powers of q where psi(), f() are Ramanujan theta functions.
L.g.f.: Log( f(-q^3, -q^5)^2 / psi(-q) ) in powers of q where psi(), f() are Ramanujan theta functions.
(End)
a(n) == 1 (mod 2) iff n is a square or twice square (A028982).
a(n) = -sigma(n) + [Sum_{d|n, d==2 (mod 4)} d] + [Sum_{d|n, d==1,4,7 (mod 8)} 2*d].