A262007 G.f.: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n / (1 - x)^n.

1, 2, 1, 8, 7, 27, 45, 102, 194, 439, 844, 1775, 3608, 7342, 14891, 30283, 61113, 123625, 249355, 502430, 1011305, 2034028, 4086860, 8206874, 16469851, 33035697, 66234208, 132746099, 265961186, 532718115, 1066778721, 2135822309, 4275459594, 8557335615, 17125445126, 34268966022, 68568212859, 137187104632

Offset: 1

Author

Paul D. Hanna, Sep 21 2015

Keywords

• nonn

Comments

Compare to the curious identity: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.

Compare also to the g.f. of A077229, where A077229(n) equals the number of compositions of n where the largest part is <= the number of parts.

Examples

			G.f.: A(x) = x + 2*x^2 + x^3 + 8*x^4 + 7*x^5 + 27*x^6 + 45*x^7 + 102*x^8 + 194*x^9 + 439*x^10 + 844*x^11 + 1775*x^12 +...
such that A(x) = N(x) + P(x) where
N(x) = Sum_{n>=1} (-1)^n * x^(n^2-n) * (1 - x)^n / (1 - x^n)^n
P(x) = Sum_{n>=0} x^n * (1 - x^n)^n / (1 - x)^n.
Explicitly,
N(x) = -1 + x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 4*x^6 - 3*x^7 + 4*x^8 - 10*x^9 + 18*x^10 - 19*x^11 + 9*x^12 + 2*x^13 + x^14 - 22*x^15 + 50*x^16 +...
P(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 11*x^5 + 23*x^6 + 48*x^7 + 98*x^8 + 204*x^9 + 421*x^10 + 863*x^11 + 1766*x^12 + 3606*x^13 + 7341*x^14 + 14913*x^15 + 30233*x^16 +...+ A077229(n)*x^n +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A077229, A260147.

Programs

PARI

{a(n) = local(A=1);
A = sum(k=-n-1, n+1, x^k * (1-x^k)^k / (1-x +x*O(x^n))^k); polcoeff(A, n)}
for(n=1, 31, print1(a(n), ", "))

PARI

{a(n) = local(A=1);
A = sum(k=-n-1, n+1, (-1)^k * x^(k^2-k) * (1 - x)^k / (1 - x^k +x*O(x^n))^k); polcoeff(A, n)}
for(n=1, 31, print1(a(n), ", "))

Formula

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

Limit a(n)^(1/n) = 2.

a(n) ~ 2^(n-1). - Vaclav Kotesovec, Sep 03 2017