0,3

Partial sum of first n harmonic numbers multiplied by n!: a(n) = n!*Sum[Sum[1/k,{k,1,m}],{m,1,n}] = n!*Sum[H(m),{m,1,n}], whrere H(m) = Sum[1/k,{k,1,m}] = A001008(m)/A002805(m) is m-th Harmonic number.

In the symmetric group S_n, each permutation factors into k independent cycles; a(n) = sum k over S_n. - Harley Flanders (harley(AT)umich.edu), Jun 28 2004

Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 22 2008: (Start)

a(n) is also the sum of the positions of the right-to-left minima in all permutations of [n]. Example: a(3)=26 because the positions of tle right-to-left minima in the permutations 123,132,213,231,312 and 321 are 123, 13, 23, 23, 3 and 3, respectively and 1+2+3+1+3+2+3+2+3+3+3=26.

a(n)=Sum(k*A143947(n,k),k=n..n(n+1)/2).

(End)

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 20 2009: (Start)

The asymptotic expansion of the higher order exponential integral E(x,m=2,n=2) ~ exp(-x)/x^2*(1 - 5/x + 26/x^2 - 154/x^3 + 1044/x^4 - 8028/x^5 + 69264/x^6 - ...) leads to the sequence given above. See A163931 and A028421 for more information.

(End)

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).

Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 406

E.g.f.: - ln ( 1 - x ) / ( 1 - x )^2. a(n) = (n+1)! * H[ n ] - n*n!, H[ n ] = sum[ k=1..n ] k^-1.

a(n) = a(n-1)*(n+1)+n! = A000254(n+1)-A000142(n+1) = A067176(n+1, 1) - Henry Bottomley (se16(AT)btinternet.com), Jan 09 2002

a(n)=sum((-1)^(n-1+k)*(k+1)*2^k*stirling1(n, k+1), k=0..n-1). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

With alternating signs: Ramanujan polynomials psi_2(n, x) evaluated at 0. - Ralf Stephan, Apr 16 2004

a(n) = n!*Sum[(k+1)/(n-k), {k, 0, n-1}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 09 2004. Examples: a(6) = 6!*(1/6+2/5+3/4+4/3+5/2+6/1) = 8028; a(20) = 20!*(1/20+2/19+3/18+4/17+5/16+...+16/5+17/4+18/3+19/2+20/1) = 135153868608460800000.

a(n) = Sum[k StirlingCycle[n+1,k+1],{k,1,n}]. - David Callan (callan(AT)stat.wisc.edu), Sep 25 2006

For n>=1, a(n)=sum((-1)^(n-j-1)*2^j*(j+1)*stirling1(n,j+1),j=0..n-1); [From Milan R. Janjic (agnus(AT)blic.net), Dec 14 2008]

(1-x)^-2 * (-log(1-x)) = x + 5/2*x^2 + 13/3*x^3 + 77/12*x^4 + ...

Examples: a(6) = 6!*(1/6+2/5+3/4+4/3+5/2+6/1) = 8028; a(20) = 20!*(1/20+2/19+3/18+4/17+5/16+...+16/5+17/4+18/3+19/2+20/1) = 135153868608460800000

a:=n->sum(n!/k, k=2..n): seq(a(n), n=1..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 22 2008

Table[n!*Sum[Sum[1/k, {k, 1, m}], {m, 1, n}], {n, 0, 20}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 14 2006

Cf. A000254, A006675.

a(n)=A112486(n, 1).

Cf. A001008, A002805.

A143947 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 22 2008]

Sequence in context: A045379 A053487 A082029 this_sequence A081047 A057793 A090226

Adjacent sequences: A001702 A001703 A001704 this_sequence A001706 A001707 A001708

nonn,easy

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

More terms from Sascha Kurz (sascha.kurz(AT)uni-bayreuth.de), Mar 22 2002

Additional comments from Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 09 2004