0,3

This sequence shares divisibility properties with A000522; each of the primes in A072456 divide only a finite number of terms of this sequence. - T. D. Noe (noe(AT)sspectra.com), Jul 07 2005

Sum of the lengths of the first runs in all permutations of [n]. Example: a(3)=10 because the lengths of the first runs in the permutation (123),(13)2,(3)12,(2)13,(23)1 and (3)21 are 3,2,1,1,2 and 1, respectively (first runs are enclosed between parentheses). Number of cells in the last columns of all deco polyominoes of height n. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. a(n)=Sum(k*A092582(n,k), k=1..n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 16 2006

As the formula a(n)=n!*Sum(1/k!, k=1..n) suggests, the terms of this sequence starting at n=1 are numerators of the fractions Sum(1/k!, k=1..n). - Alexander R. Povolotsky (pevnev(AT)juno.com), Dec 02 2007

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 27 2009: (Start)

Starting with offset 1 = eigensequence of an infinite lower triangular matrix

with (1, 2, 3,...) as the right border, (1, 1, 1,...) as the left border, and the rest zeros. (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).

D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70.

E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

T. D. Noe, Table of n, a(n) for n=0..100

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 150

a(n)=n!*Sum(1/k!, k=1..n).

a(n) = floor [ {(n!)* (e -1)}] - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 08 2002

E.g.f.: (e^z-1)/(1-z) - Mario Catalani (mario.catalani(AT)unito.it), Feb 06 2003

Binomial transform of A002467. - Ross La Haye (rlahaye(AT)new.rr.com), Sep 21 2004

a:=n->sum((n-j)!*binomial(n,j),j=1..n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 31 2006

a(n)=1+sum(k=0,n-1,k*a(k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jul 26 2008

a(m)=int(((1+s)^m-s^m)*exp(-s),s=0..infinity)=GAMMA(m+1,1)*exp(1)-GAMMA(m+1) [From Stephen Crowley (crow(AT)crowlogic.net), Jul 24 2009]

[a(0),a(1),..]=GAMMA(m+1,1)*exp(1)-GAMMA(m+1)=[exp(-1)*exp(1)-1, 2*exp(-1)*exp(1)-1, 5*exp(-1)*exp(1)-2, 16*exp(-1)*exp(1)-6, 65*exp(-1)*exp(1)-24, 326*exp(-1)*exp(1)-120,...] [From Stephen Crowley (crow(AT)crowlogic.net), Jul 24 2009]

a:=n->sum((n-j)!*binomial(n, j), j=1..n): seq(a(n), n=0..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 31 2006

FoldList[ #1*#2 + 1 &, 0, Range[21]] (from Robert G. Wilson v (rgwv(at)rgwv.com), Oct 11 2005)

A002627(n) = A000522(n) - n!. Second diagonal of A059922, Cf. A056542.

Cf. A092582.

Sequence in context: A116540 A000248 A030927 this_sequence A030802 A030942 A030855

Adjacent sequences: A002624 A002625 A002626 this_sequence A002628 A002629 A002630

nonn,easy,nice

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

Comments from Michael Somos

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 16 2006