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.

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.

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

[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.]

Row sums of triangle A026998.

Cf. A000919, A001117, A001118.

Cf. A095121. A110146.

Adjacent sequences: A000915 A000916 A000917 this_sequence A000919 A000920 A000921

Sequence in context: A063452 A009299 A072611 this_sequence A095121 A122958 A122959

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