A292180 G.f.: Sum_{n=-oo..+oo} (1 + x^n)^n / (1 - x^n)^n, ignoring the constant term.

4, 0, 16, 16, 24, 0, 32, 96, 116, 0, 48, 192, 56, 0, 608, 704, 72, 0, 80, 480, 1408, 0, 96, 3712, 2108, 0, 2720, 896, 120, 0, 128, 9600, 4672, 0, 17088, 12112, 152, 0, 7392, 20800, 168, 0, 176, 2112, 63032, 0, 192, 134400, 57828, 0, 15648, 2912, 216, 0, 130336, 69888, 21440, 0, 240, 317056, 248, 0, 556960, 428800, 282576, 0, 272, 4896, 36992, 0, 288, 1029600, 296, 0, 599024, 6080, 1859712, 0

Offset: 1

Author

Paul D. Hanna, Sep 24 2017

Keywords

• nonn

Comments

a(4*n-2) = 0 for n>=1.

a(n) is divisible by 4 for n>=1.

a((2*n-1)^2)/4 is odd for n>=1 (conjecture).

Examples

			G.f.: A(x) = 4*x + 16*x^3 + 16*x^4 + 24*x^5 + 32*x^7 + 96*x^8 + 116*x^9 + 48*x^11 + 192*x^12 + 56*x^13 + 608*x^15 + 704*x^16 + 72*x^17 + 80*x^19 + 480*x^20 + 1408*x^21 + 96*x^23 + 3712*x^24 + 2108*x^25 + 2720*x^27 + 896*x^28 + 120*x^29 + 128*x^31 + 9600*x^32 + 4672*x^33 + 17088*x^35 + 12112*x^36 + 152*x^37 + 7392*x^39 + 20800*x^40 +...
where A(x) = Sum_{n=-oo..+oo} (1 + x^n)^n / (1 - x^n)^n, ignoring constant terms.
G.f. A(x) = P(x) + Q(x), where
P(x) = Sum_{n>=1} (1 + x^n)^n / (1 - x^n)^n - 1,
explicitly,
P(x) = 2*x + 6*x^2 + 8*x^3 + 18*x^4 + 12*x^5 + 44*x^6 + 16*x^7 + 66*x^8 + 58*x^9 + 92*x^10 + 24*x^11 + 276*x^12 + 28*x^13 + 156*x^14 + 304*x^15 + 386*x^16 + 36*x^17 + 674*x^18 + 40*x^19 + 1092*x^20 + 704*x^21 + 332*x^22 + 48*x^23 + 2852*x^24 + 1054*x^25 + 444*x^26 + 1360*x^27 + 3124*x^28 + 60*x^29 + 6648*x^30 + 64*x^31 + 4866*x^32 + 2336*x^33 + 716*x^34 + 8544*x^35 + 15494*x^36 + 76*x^37 + 876*x^38 + 3696*x^39 + 25796*x^40 +...
and
Q(x) = Sum_{n>=1} (-1)^n * (1 - x^n)^n / (1 + x^n)^n - (-1)^n,
explicitly,
Q(x) = 2*x - 6*x^2 + 8*x^3 - 2*x^4 + 12*x^5 - 44*x^6 + 16*x^7 + 30*x^8 + 58*x^9 - 92*x^10 + 24*x^11 - 84*x^12 + 28*x^13 - 156*x^14 + 304*x^15 + 318*x^16 + 36*x^17 - 674*x^18 + 40*x^19 - 612*x^20 + 704*x^21 - 332*x^22 + 48*x^23 + 860*x^24 + 1054*x^25 - 444*x^26 + 1360*x^27 - 2228*x^28 + 60*x^29 - 6648*x^30 + 64*x^31 + 4734*x^32 + 2336*x^33 - 716*x^34 + 8544*x^35 - 3382*x^36 + 76*x^37 - 876*x^38 + 3696*x^39 - 4996*x^40 +...
Terms at square positions divided by 4 begin:
a(n^2)/4 = [1, 4, 29, 176, 527, 3028, 14457, 107200, 446745, 2392604, 13286165, 140564336, 415382567, 2333455268, 17078911507, 78663453440, 419472490547, 2377516612900, 13482186743565, 78663154105296, 437169506932981, 2481447593907572, 14146164790774889, 161511806183206336, 460995825168188653, 2634869356953946428, 15071070681878977525, 86632929673574593072, 494051395886263605335, 2955861929786748934348, 16234283204352299108321, ...].
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..1024

Crossrefs

Cf. A292180, A261608.

Programs

PARI

{a(n) = my(A,Ox=x*O(x^n)); A = sum(n=-n-1,n+1, if(n==0,0, (1 + x^n +Ox)^n/(1-x^n +Ox)^n - 1/2 +Ox )); polcoeff(A,n)}
for(n=1,80,print1(a(n),", "))

PARI

{a(n) = my(A,Ox=x*O(x^n)); A = sum(m=1,n+1, ((1+x^m +Ox)^(2*m) + (-1)^m*(1 - x^m +Ox)^(2*m))/(1 - x^(2*m) +Ox)^m - 1 ); polcoeff(A,n)}
for(n=1,80,print1(a(n),", "))

Formula

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