0,3

For n>2, sum(k=1,a(n),(-1)^C(n,k) ) = A064405(a(n))+1 = 0 - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 18 2002

For n > 0, the number of nonempty proper subsets of an n element set. - Ross La Haye (rlahaye(AT)new.rr.com), Feb 07 2004

Numbers n such that abs( sum(k=0,n,(-1)^C(n,k)*C(n+k,n-k)) ) = 1 - Benoit Cloitre (benoit7848c(AT)orange.fr), Jul 03 2004

For n>2 this formula also counts edge rooted forests in a cycle of length n. - Woong Kook (andrewk(AT)math.uri.edu), Sep 08 2004

For n >= 1, conjectured to be the number of integers from 0 to (10^n)-1 that lack 0, 1, 2, 3, 4, 5, 6 and 7 as a digit. - Alexandre Wajnberg (alexandre.wajnberg(AT)ulb.ac.be), Apr 25 2005

Beginning with a(2)=2, these are the partial sums of the subsequence of A000079=2^n beginning with A000079(1)=2. Hence for n >= 2 a(n) is the smallest possible sum of exactly one prime, one two-almost prime, one three-almost prime, ... and one (n-1)-almost prime. A060389 (partial sums of the primorials, A002110, beginning with A002110(1)=2) is the analogue when all the almost primes must also be squarefree. - Rick L. Shepherd (rshepherd2(AT)hotmail.com), May 20 2005

From the second term on (n>=1), the binary representation of these numbers is a 0 preceded by (n-1) 1's. This pattern (0)111...1110 is the "opposite" of the binary 2^n+1: 1000...0001 (cf. A000051). - Alexandre Wajnberg (alexandre.wajnberg(AT)ulb.ac.be), May 31 2005

The numbers 2^n-2 (n>=2) give the positions of 0's in A110146. Also numbers n such that n^(n+1) = 0 mod (n+2). - Zak Seidov (zakseidov(AT)yahoo.com), Feb 20 2006

Number of surjections from an n-element set onto a two-element set, with n >= 2. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Dec 15 2007

It appears that these are the numbers n such that 3*A135013(n) = n*(n+1), thus answering Problem 2 on the Mathematical Olympiad Foundation of Japan, Final Round Problems, Feb 11 1993.

a(n) = A058896(n)/A052548(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 14 2009]

Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if x is a proper subset of y or y is a proper subset of x and x and y are disjoint. Then a(n+1) = |R|. [From Ross La Haye (rlahaye(AT)new.rr.com), Mar 19 2009]

a(n) = A164874(n-1,n-1) for n>1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 29 2009]

Apart the first term which is -1 the number of units of a(n) belongs to a periodic sequence: 0, 2, 6, 4. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.

Mathematical Olympiad Foundation of Japan, Final Round Problems, Feb 11 1993, Problem 2.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.

A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.

Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. [From Ross La Haye (rlahaye(AT)new.rr.com), Mar 19 2009]

Index entries for sequences related to linear recurrences with constant coefficients

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 77

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Eric Weisstein's World of Mathematics, Sphere Line Picking

G.f.: 1/(1-2x) - 2/(1-x), e.g.f.: (e^x - 1)^2 - 1. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001

For n>=1, a(n) = A008970(n+1, 2) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 21 2004

G.f.: (3x - 1)/(2x^2 - 3x + 1).

a(n) = 2a(n-1) + 2 - Alexandre Wajnberg (alexandre.wajnberg(AT)ulb.ac.be), Apr 25 2005

a(n) = A000079(n)-2. [From Omar E. Pol (info(AT)polprimos.com), Dec 16 2008]

[seq (stirling2(n, 2)*2, n=0..28)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 06 2006

ZL := [S, {S=Prod(B, B), B=Set(Z, 1 <= card)}, labeled]: seq(combstruct[count](ZL, size=n), n=0..28); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 13 2007

a:=n->sum (2^j, j=1..n): seq(a(n), n=-1..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007

A000918:=2*z/((2*z-1)*(z-1)); [S. Plouffe in his 1992 dissertation.]

lst={}; Do[AppendTo[lst, 2^n-2], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 16 2008]

(Other) sage: [gaussian_binomial(n, 1, 2)-1 for n in xrange(0, 29)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 31 2009]

Row sums of triangle A026998.

Cf. A000919, A001117, A001118.

Cf. A095121. A110146.

Sequence in context: A063452 A009299 A072611 this_sequence A095121 A122958 A122959

Adjacent sequences: A000915 A000916 A000917 this_sequence A000919 A000920 A000921

sign,easy

N. J. A. Sloane (njas(AT)research.att.com).