A293602 G.f.: Product_{n=-oo..+oo} ( 1 + x^n*(1 - x^n)^n ).

-2, -4, -6, -12, -14, -22, -30, -40, -38, -44, -64, -54, -40, 16, 12, 54, 94, 248, 370, 672, 754, 932, 1112, 1360, 2388, 2684, 3508, 4388, 4452, 4240, 7342, 7538, 9852, 14268, 19268, 16892, 22950, 24514, 24854, 41138, 50750, 54974, 89336, 117050, 103586, 140074, 153180, 147910, 236252, 267510, 274156, 520150, 515664, 568144, 906496, 834244, 552372, 1130104, 1142566, 933698, 1772950, 1915700

Offset

1,1

Comments

Compare g.f. to: Sum_{n=-oo..+oo} x^n*(1 - x^n)^n = 0.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..520

Formula

G.f.: A(x) = P(x) * Q(x) where

P(x) = Product_{n>=0} ( 1 + x^n*(1 - x^n)^n ),

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

Example

G.f.: A(x) = -2*x - 4*x^2 - 6*x^3 - 12*x^4 - 14*x^5 - 22*x^6 - 30*x^7 - 40*x^8 - 38*x^9 - 44*x^10 - 64*x^11 - 54*x^12 - 40*x^13 + 16*x^14 + 12*x^15 + 54*x^16 + 94*x^17 + 248*x^18 + 370*x^19 + 672*x^20 + 754*x^21 + 932*x^22 + 1112*x^23 + 1360*x^24 + 2388*x^25 + 2684*x^26 + 3508*x^27 + 4388*x^28 + 4452*x^29 + 4240*x^30 +...
where
A(x) = P(x) * Q(x) where
P(x) = 2 * (1 + x*(1-x)) * (1 + x^2*(1-x^2)^2) * (1 + x^3*(1-x^3)^3) * (1 + x^4*(1-x^4)^4) * (1 + x^5*(1-x^5)^5) *...* ( 1 + x^n*(1 - x^n)^n ) *...
Q(x) = (1 - 1/(1-x)) * (1 + x^2/(1-x^2)^2) * (1 - x^6/(1-x^3)^3) * (1 + x^12/(1-x^4)^4) * (1 - x^20/(1-x^5)^5) *...* (1 + (-1)^n * x^(n^2-n)/(1 - x^n)^n ) *...
Explicitly,
P(x) = 2 + 2*x + 4*x^3 - 2*x^4 + 6*x^6 - 2*x^7 - 10*x^8 + 22*x^10 - 18*x^11 - 36*x^13 - 4*x^14 + 98*x^15 - 10*x^16 - 104*x^17 + 30*x^18 - 108*x^19 - 40*x^20 + 448*x^21 - 16*x^22 - 200*x^23 - 214*x^24 - 148*x^25 - 586*x^26 + 718*x^27 + 1374*x^28 - 224*x^29 + 104*x^30 +...
Q(x) = -x - x^2 - 2*x^3 - 2*x^4 - 4*x^5 - 4*x^6 - 6*x^7 - 6*x^8 - 9*x^9 - 6*x^10 - 9*x^11 - 6*x^12 - 4*x^13 + 2*x^14 + 4*x^15 + 23*x^16 + 26*x^17 + 48*x^18 + 72*x^19 + 107*x^20 + 123*x^21 + 195*x^22 + 232*x^23 + 317*x^24 + 376*x^25 + 515*x^26 + 595*x^27 + 817*x^28 + 912*x^29 + 1215*x^30 +...

Mathematica

terms = 62; Product[If[n == 0, 2, (1 + x^n*(1 - x^n)^n)], {n, -terms, terms}] + O[x]^(terms+1) // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Nov 04 2017 *)

Prog

(PARI) {a(n) = my(A); A = prod(m=-n-1, n+1, (1 + x^m*(1 - x^m)^m + x*O(x^n)) ); polcoeff(A, n)}
for(n=1, 60, print1(a(n), ", "))

Crossrefs

Cf. A293603, A295131.

Sequence in context: A331620 A161337 A050055 * A015632 A111084 A015636

Adjacent sequences: A293599 A293600 A293601 * A293603 A293604 A293605

Keyword

sign,look

Author

Paul D. Hanna, Oct 21 2017

Status

approved