0,5

Number of palindromic structures using exactly three different symbols. - Marks R. Nester (nesterm(AT)dpi.qld.gov.au)

Number of ways of placing n labeled balls into k=3 indistinguishable boxes. - Thomas Wieder (wieder.thomas(AT)t-online.de), Nov 30 2004

With two leading zeros, this is the second binomial transform of cosh(x)-1, and the binomial transform of A000225 (with extra leading zero). - Paul Barry (pbarry(AT)wit.ie), May 13 2003

Let [m] denote the first m positive integers. Then a(n) is the number of functions f from [n] to [n+1] that satisfy (i) f(x)>x for all x, (ii) f(x)=n+1 for exactly 3 elements, and (iii) f(f(x))=n+1 for the remaining n-3 elements of [n]. For example, a(4)=6 since there are exactly 6 functions from {1,2,3,4} to {1,2,3,4,5} such that f(x) > x, f(x) = 5 for 3 elements, and f(f(x)) = 5 for the remaining element. The functions are f1 = {(1,5), (2,5), (3,4), (4,5)}, f2 = {(1,5), (2,3), (3,5), (4,5)}, f3 = {(1,5), (2,4), (3,5), (4,5)}, f4 = {(1,2), (2,5), (3,5), (4,5)}, f5 = {(1,3), (2,5), (3,5), (4,5)}, f6 = {(1,4), (2,5), (3,5), (4,5)}. - Dennis P. Walsh (dwalsh(AT)mtsu.edu), Feb 20 2007

Conjecture. Let S(1)={1} and, for n>1, let S(n) be the smallest set containing x, 2x and 3x for each element x in S(n-1). Then a(n) is the sum of the elements in S(n). (It is easy to prove that the number of elements in S(n) is the nth triangular number given by A001952.) See A122554 for a sequence defined in this way. - John W. Layman (layman(AT)math.vt.edu), Nov 21 2007

Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which x and y are disjoint and for which x is not a subset of y and y is not a subset of x. Wieder calls these "disjoint strict k-combinations". - Ross La Haye (rlahaye(AT)new.rr.com), Jan 11 2008

Also, let P(A) be the power set of an n-element set A. Then a(n+2) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 2) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x. - Ross La Haye (rlahaye(AT)new.rr.com), Jan 11 2008

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia.

T. D. Noe, Table of n, a(n) for n=0..200

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 346

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, Link to a section of The World of Mathematics.

Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).

Zerinvary Lajos, Sage Notebooks

G.f.: x^3/((1-x)*(1-2*x)*(1-3*x)). E.g.f.: ((exp(x)-1)^3)/3!.

Recurrence: a(n+3) = 6a(n+2) - 11a(n+1) + 6(n), a(3) = 1, a(4) = 6, a(5) = 25. - Thomas Wieder (wieder.thomas(AT)t-online.de), Nov 30 2004

With offset 0, this is 9*3^n/2-4*2^n+1/2, the partial sums of 3*3^n-2*2^n=A001047(n+1) - Paul Barry (pbarry(AT)wit.ie), Jun 26 2003

a(n)=(1+3^(n-1)-2^n)/2 - Dennis P. Walsh (dwalsh(AT)mtsu.edu), Feb 20 2007

a(n)=(3^n-1)/2-(2^n-1), n>=2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007

a(4) = 6. Let denote Z[i] the i-th labeled element = "ball". Then one has for n=4 six different ways to fill sets = "boxes" with the labeled elements:

Set(Set(Z[3], Z[4]), Set(Z[1]), Set(Z[2])), Set(Set(Z[3], Z[1]), Set(Z[4]), Set(Z[2])), Set(Set(Z[4], Z[1]), Set(Z[3]), Set(Z[2])), Set(Set(Z[4]), Set(Z[1]), Set(Z[3], Z[2])), Set(Set(Z[3]), Set(Z[1], Z[2]), Set(Z[4])), Set(Set(Z[3]), Set(Z[1]), Set(Z[4], Z[2]))

A000392 := n -> 9/2*3^n-4*2^n+1/2; [ seq(9/2*3^n-4*2^n+1/2, n=0..30) ]; (from Thomas Wieder)

a:=n->sum(3^(n-j)-2^(n-j), j=0..n): seq(a(n), n=1..25); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 04 2007

seq((3^n-1)/2-(2^n-1), n=2..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007

A000392:=-1/(z-1)/(3*z-1)/(2*z-1); [S. Plouffe in his 1992 dissertation.]

(PARI) a(n)=3^(n-1)/2-2^(n-1)+1/2

sage: [stirling_number2(i, 3) for i in xrange(0, 40)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 26 2008

Cf. A008277 (Stirling2 triangle), A007051, A056509, A000225.

Cf., A003462, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A016256.

Cf. A028243, A122554.

Sequence in context: A056279 A055337 A001871 this_sequence A099948 A092491 A112308

Adjacent sequences: A000389 A000390 A000391 this_sequence A000393 A000394 A000395

nonn,nice,easy

njas

Offset changed by njas, Feb 08 2008