1,4

a(n+1) = Sum_{pi in Symm(n)} Sum_{i=1..n} max(pi(i)-i,0)^2, i.e. the sum of the squares of the positive displacement of all letters in all permutations on n letters. - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Oct 25 2006

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).

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.

E.g.f.: ((x/(1-x))^3)/3!.

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*f(n,2,-4), (n>=2). [From Milan R. Janjic (agnus(AT)blic.net), Mar 01 2009]

[seq(n!*binomial(n-1, 2)/6, n=1..40)];

a:=n->sum((n-j)*n!/6, j=2..n): seq(a(n), n=1..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 29 2007

restart: G(x):=x^3/(1-x)^3: 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=1..21); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 01 2009]

Column 3 of A008297. Cf. A053495.

Column m=3 of unsigned triangle A111596.

Cf. A005990.

Sequence in context: A129329 A129332 A004291 this_sequence A037511 A037694 A067102

Adjacent sequences: A001751 A001752 A001753 this_sequence A001755 A001756 A001757

nonn

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