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A377099
G.f. A(x) satisfies A(x^3)/A(x)^3 = 1 - 3*Sum_{n>=1} x^n/(1 + x^n + x^(2*n)).
1
1, 1, 2, 6, 17, 49, 146, 444, 1366, 4243, 13276, 41818, 132425, 421186, 1344724, 4307464, 13837365, 44563750, 143841132, 465214791, 1507319118, 4891734608, 15898701393, 51742236543, 168602613351, 550016490269, 1796144224799, 5871199729958, 19208895229376, 62898558771219, 206118700436344
OFFSET
0,3
COMMENTS
Compare to an identity of the Jacobi theta_4 function (see the Borwein link),
theta_4(q)^3/theta_4(q^3) = 1 - 6*Sum_{n>=0} q^(3*n+1)/(1 + q^(3*n+1)) - q^(3*n+2)/(1 + q^(3*n+2)).
Related identity: Sum_{n>=1} (-x)^n/(1 + x^n + x^(2*n)) = Sum_{n>=0} x^(3*n+1)/(1 + x^(3*n+1)) - x^(3*n+2)/(1 + x^(3*n+2)).
Related sequence: A002324(n) = the number of divisors of n == 1 (mod 3) minus number of divisors of n == 2 (mod 3).
LINKS
J. M. Borwein, P. B. Borwein and F. Garvan, Some Cubic Modular Identities of Ramanujan, Trans. Amer. Math. Soc. 343 (1994), p. 44.
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x^3)/A(x)^3 = 1 - 3*Sum_{n>=1} x^n/(1 + x^n + x^(2*n)).
(2) ( A(x^3)/A(x)^3 - A(x^9)/A(x^3)^3 ) * ( A(x^9)/A(x^3)^3 - A(x^18)/A(x^6)^3 ) = ( A(x^6)/A(x^2)^3 - A(x^18)/A(x^6)^3 )^2. See formula by Michael Somos in A002324 dated May 20 2005.
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 17*x^4 + 49*x^5 + 146*x^6 + 444*x^7 + 1366*x^8 + 4243*x^9 + 13276*x^10 + 41818*x^11 + 132425*x^12 + ...
RELATED SERIES.
A(x^3)/A(x)^3 = 1 - 3*x - 3*x^3 - 3*x^4 - 6*x^7 - 3*x^9 - 3*x^12 - 6*x^13 - 3*x^16 - 6*x^19 - 6*x^21 - 3*x^25 + ... + (-3)*A002324(n)*x^n + ...
Sum_{n>=1} x^n/(1 + x^n + x^(2*n)) = x + x^3 + x^4 + 2*x^7 + x^9 + x^12 + 2*x^13 + x^16 + 2*x^19 + 2*x^21 + x^25 + ... + A002324(n)*x^n + ...
From the series expansion of 1/A(x)^3,
1/A(x)^3 = 1 - 3*x - 4*x^3 + 2*x^6 - 3*x^9 + 18*x^10 + 3*x^12 + 27*x^13 + 10*x^15 + 81*x^16 + 18*x^18 + 189*x^19 + 31*x^21 + 513*x^22 + 87*x^24 + 1377*x^25 + 194*x^27 + 3858*x^28 + 508*x^30 + ...
we see that the coefficient of x^(3*n+2) in 1/A(x)^3 is zero for n >= 0.
SPECIFIC VALUES.
A(t) = 2 at t = 0.26663565103127589661372886956252964388199649379033...
A(1/4) = 1.7456273242832904615846006230512476127224338964290...
A(1/5) = 1.3952978173469146622650699069014967276370657853776...
A(1/6) = 1.2758332458375016728011403530884129499997540675161...
A(1/10) = 1.128400243120363122917130283440909306570669657292...
PROG
(PARI) {a(n) = my(V=[1], A); for(i=0, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef( subst(A, x, x^3)/A^3 - 1 + 3*sum(n=1, #V, x^n/(1 + x^n + x^(2*n) +x*O(x^#V))), #V-1)/3 ); V[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Sequence in context: A122100 A122099 A026165 * A336742 A148445 A148446
KEYWORD
nonn,changed
AUTHOR
Paul D. Hanna, Oct 19 2024
STATUS
approved