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%I #14 Sep 03 2017 07:22:27
%S 1,2,1,8,7,27,45,102,194,439,844,1775,3608,7342,14891,30283,61113,
%T 123625,249355,502430,1011305,2034028,4086860,8206874,16469851,
%U 33035697,66234208,132746099,265961186,532718115,1066778721,2135822309,4275459594,8557335615,17125445126,34268966022,68568212859,137187104632
%N G.f.: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n / (1 - x)^n.
%C Compare to the curious identity: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.
%C 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.
%H Paul D. Hanna, <a href="/A262007/b262007.txt">Table of n, a(n) for n = 1..1000</a>
%F G.f.: Sum_{n=-oo..+oo} (-1)^n * x^(n^2-n) * (1 - x)^n / (1 - x^n)^n.
%F Limit a(n)^(1/n) = 2.
%F a(n) ~ 2^(n-1). - _Vaclav Kotesovec_, Sep 03 2017
%e 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 +...
%e such that A(x) = N(x) + P(x) where
%e N(x) = Sum_{n>=1} (-1)^n * x^(n^2-n) * (1 - x)^n / (1 - x^n)^n
%e P(x) = Sum_{n>=0} x^n * (1 - x^n)^n / (1 - x)^n.
%e Explicitly,
%e 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 +...
%e 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 +...
%o (PARI) {a(n) = local(A=1);
%o A = sum(k=-n-1, n+1, x^k * (1-x^k)^k / (1-x +x*O(x^n))^k); polcoeff(A, n)}
%o for(n=1, 31, print1(a(n), ", "))
%o (PARI) {a(n) = local(A=1);
%o 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)}
%o for(n=1, 31, print1(a(n), ", "))
%Y Cf. A077229, A260147.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Sep 21 2015
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