1,3

Coefficients of Gandhi polynomials.

a(n) = Sum_{pi in Symm(n)} Sum_{i=1..n} max(pi(i)-i,0), i.e. the total positive displacement of all letters in all permutations on n letters. - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Oct 25 2006

a(n) is also the sum of the excedances of all permutations of [n]. An excedance of a permutation p of [n] is an i (1<=i<=n-1) such that p(i)>i. Proof: i is an excedance if p(i)=i+1, i+2, ..., n (n-i possibilities), with the remaining values of p forming any permutation of [n]\{p(i)} in the positions [n]\{i} ((n-1)! possibilities). Summation of i(n-i)(n-1)! over i from 1 to n-1 completes the proof. Example: a(3)=8 because the permutations 123, 132, 213, 231, 312, 321 have excedances NONE, {2}, {1}, {1,2}, {1}, {1}, respectively. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 26 2008]

Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 26 2009: (Start)

a(n) is also the number of doubledescents in all permutations of {1,2,...,n-1}. We say that i is a doubledescent of a permutation p if p(i)>p(i+1)>p(i+2). Example: a(3)=8 because each of the permutations 1432, 4312, 4213, 2431, 3214, 3421 has one doubledescent, the permutation 4321 has two doubledescents and the remaining 17 permutations of {1,2,3,4} have no doubledescents.

(End)

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

D. Dumont, Interpretations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

a(n)=A052571(n+2)/6. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2007

a(n)=sum(sum(sum(n!/6, j=1..n),k=-1..n),m=0..n), n>=0 . - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2007

If we define f(n,i,x)= sum(sum(binomial(k,j)*stirling1(n,k)*stirling2(j,i)*x^(k-j),j=i..k),k=i..n) then a(n+1)=(-1)^(n-1)*f(n,1,-4), (n>=1). [From Milan R. Janjic (agnus(AT)blic.net), Mar 01 2009]

[ seq((n-1)*(n+1)!/6, n=1..40) ];

a:=n->sum(sum(sum(n!/6, j=1..n), k=-1..n), m=0..n): seq(a(n), n=0..19); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2007

seq(sum(mul(j, j=3..n), k=3..n)/3, n=2..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 01 2007

restart: G(x):=x^3/(1-x)^2: f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n]/3!, n=2..21); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 01 2009]

Table[Sum[n!/6, {i, 3, n}], {n, 2, 21}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 12 2009]

A090672(n)/2.

Cf. A001715.

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 12 2009: (Start)

Equals the second right hand column of A167568 divided by 2.

(End)

Sequence in context: A001267 A099156 A129331 this_sequence A160228 A099337 A075147

Adjacent sequences: A005987 A005988 A005989 this_sequence A005991 A005992 A005993

nonn,easy

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

Formula from Robert Newstedt.