A380680 G.f. A(x) = Sum_{n=-oo..+oo} x^(n^2) * (2 - x^n)^n/(1 - 2*x^n)^n.

1, 4, 6, 12, 32, 48, 120, 192, 450, 784, 1704, 3072, 6624, 12288, 25536, 49404, 100544, 196608, 398982, 786432, 1584192, 3147924, 6316032, 12582912, 25226304, 50331712, 100780032, 201341856, 402908016, 805306368, 1611192192, 3221225472, 6443615490, 12884995776, 25772261376, 51539609952

Offset

0,2

Comments

The term x^(n^2) * (2 - x^n)^n/(1 - 2*x^n)^n is invariant under sign reversal of n.

Conjecture: for n > 0, a(n) == 2 (mod 4) iff n is a square, and a(n) is divisible by 4 when n is nonsquare.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1030

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) A(x) = Sum_{n=-oo..+oo} x^(n^2) * (2 - x^n)^n/(1 - 2*x^n)^n.

(2) A(x) = 1 + 2*Sum_{n>=1} x^(n^2) * (2 - x^n)^n/(1 - 2*x^n)^n.

(3) A(x) = 1 + 2*Sum_{n>=1} x^(n^2) * (3*(1 - x^n)/(1 - 2*x^n) - 1)^n.

(4) A(x) = 1 + 2*Sum_{n>=1} x^(n^2) * Sum_{k=0..n} binomial(n,k) * (-1)^(n-k) * 3^k * (1 - x^n)^k/(1 - 2*x^n)^k.

Example

G.f.: A(x) = 1 + 4*x + 6*x^2 + 12*x^3 + 32*x^4 + 48*x^5 + 120*x^6 + 192*x^7 + 450*x^8 + 784*x^9 + 1704*x^10 + 3072*x^11 + 6624*x^12 + ...
where
A(x) = 1 + 2*x*(2-x)/(1-2*x) + 2*x^4*(2-x^2)^2/(1-2*x^2)^2 + 2*x^9*(2-x^3)^3/(1-2*x^3)^3 + 2*x^16*(2-x^4)^4/(1-2*x^4)^4 + 2*x^25*(2-x^5)^5/(1-2*x^5)^5 + 2*x^36*(2-x^6)^6/(1-2*x^2)^6 + ...
The initial expansions of (2 - x^n)^n/(1 - 2*x^n)^n are
(2 - x) / (1 - 2*x)   = 2 + 3*x + 6*x^2 + 12*x^3 + 24*x^4 + 48*x^5 + ...
(2-x^2)^2/(1-2*x^2)^2 = 4 + 12*x^2 + 33*x^4 + 84*x^6 + 204*x^8 + 480*x^10 + ...
(2-x^3)^3/(1-2*x^3)^3 = 8 + 36*x^3 + 126*x^6 + 387*x^9 + 1098*x^12 + ...
(2-x^4)^4/(1-2*x^4)^4 = 16 + 96*x^4 + 408*x^8 + 1464*x^12 + 4737*x^16 + ...
(2-x^5)^5/(1-2*x^5)^5 = 32 + 240*x^5 + 1200*x^10 + 4920*x^15 + 17850*x^20 + ...
(2-x^6)^6/(1-2*x^6)^6 = 64 + 576*x^6 + 3312*x^12 + 15264*x^18 + 61308*x^24 + ...
...
SPECIFIC VALUES.
A(t) = 9 at t = 0.41309144917215370217410572914655923718882894026997...
A(t) = 8 at t = 0.40262864087167595911285669946861306840919924539380...
A(t) = 7 at t = 0.38942050628726776518149654142585303887236212808857...
A(t) = 6 at t = 0.37223879132421035184926301339412105503508009826123...
A(t) = 5 at t = 0.34899082065834573759993651414902314672436357928556...
A(t) = 4 at t = 0.31579498073181227254554981514043636155773221953576...
A(t) = 3 at t = 0.26458995878679926906796698568116847796797967349133...
A(t) = 2 at t = 0.17608710001419230553458210734216394476610557820247...
A(t) = 3/2 at t = 0.104187776739808499047020481018427445490680070946...
A(1/3) = 4.47993091480224558040920659098283798031673458048938...
A(1/4) = 2.78837074661284223884399780719008718198128858128929...
A(1/5) = 2.21453250315032106849307222292881508882325930872507...
A(1/6) = 1.92339780438787607899676974163153674282377686305603...
A(1/7) = 1.74640556076057657793871160742764590625453409170778...
A(1/8) = 1.62704889453106113656597926197189890740366321749731...
A(1/9) = 1.54094865068142992850252299223381677802403214315581...

Prog

(PARI) {a(n) = my(A=1); A = 1 + 2*sum(m=1, n, x^(m^2) * (2 - x^m)^m/(1 - 2*x^m +O(x^n))^m ); polcoef(A, n)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A380060.

Sequence in context: A027070 A087785 A366772 * A081198 A275978 A242211

Adjacent sequences: A380677 A380678 A380679 * A380681 A380682 A380683

Keyword

nonn

Author

Paul D. Hanna, Feb 06 2025

Status

approved