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 A262007 G.f.: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n / (1 - x)^n. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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. LINKS Paul D. Hanna, Table of n, a(n) for n = 1..1000 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 EXAMPLE 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 +... PROG (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), ", ")) CROSSREFS Cf. A077229, A260147. Sequence in context: A105672 A338249 A214271 * A005489 A015152 A021461 Adjacent sequences:  A262004 A262005 A262006 * A262008 A262009 A262010 KEYWORD nonn AUTHOR Paul D. Hanna, Sep 21 2015 STATUS approved

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Last modified June 17 05:18 EDT 2021. Contains 345080 sequences. (Running on oeis4.)