0,2

Also the number of logically distinct strings of first order quantifiers in which n variables occur (C. S. Peirce, c. 1903). - Stephen Pollard (spollard(AT)truman.edu), Jun 07 2002

Stirling transform of A052849(n)=[2,4,12,48,240,...] is a(n)=[2,6,26,150,1082,..]. - Michael Somos Mar 04 2004

Stirling transform of A000142(n-1)=[1,1,2,6,24,...] is a(n-1)=[1,2,6,26,...]. - Michael Somos Mar 04 2004

Stirling transform of (-1)^n*A024167(n-1)=[0,1,-1,5,-14,94,...] is a(n-2)=[0,1,2,6,26,...]. - Michael Somos Mar 04 2004

The asymptotic expansion of 2*log(n)-(2^1log(1)+2^2log(2)+...+2^nlog(n))/2^n is a(1)/1/n +a(2)/2/n^2 +a(3)/3/n^3 +... - Michael Somos, Aug 22 2004

This is the sequence of cumulants of the probability distribution of the number of tails before the first head in a sequence of fair coin tosses. - Michael Hardy (hardy(AT)math.umn.edu), May 01 2005

N. G. de Bruijn, Asymptotic Methods in Analysis, Dover, 1981, p. 36.

Eric Hammer, The Calculations of Peirce's 4.453, Transactions of the Charles S. Peirce Society, Vol. 31 (1995), pp. 829-839.

D. E. Knuth, personal communication.

J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 174.

Dawidson RAZAFIMAHATOLOTRA, Number of Preorders to Compute Probability of Conflict of an Unstable Effectivity Function, Preprint, Paris School of Economics, University of Paris I, Nov 23 2007.

Charles Sanders Peirce, Collected Papers, eds. C. Hartshorne and P. Weiss, Harvard University Press, Cambridge, Vol. 4, 1933, pp. 364-365.

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 99

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Stirling Number of the Second Kind

For n>0, a(n) = 2*A000670(n).

a(n) = Sum {from k=1 to infinity} k^n/(2^k); a(n) = 1 + Sum {from j=0 to n-1} C(n, j) a(j); number of combinations of a Simplex lock having n buttons.

a(n) = round[n!/ln(2)^(n+1)] (just for n <= 15) - Henry Bottomley (se16(AT)btinternet.com), Jul 04 2000

a(n) is asymptotic to n!/log(2)^(n+1). - Benoit Cloitre, Oct 20, 2002

a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*k!*2^k. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 29 2003

E.g.f.: exp(x)/(2-exp(x)) = d/dx log(1/(2-exp(x))).

a(n) = Sum_{k = 1..n} A008292(n, k)*2^k; A008292: triangle of Eulerian numbers . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 05 2004

a(1)=1, a(n) = 2*sum(k! A008277(n-1, k), k=1..n-1) for n>1 or a(n) = sum((k-1)! A008277(n, k), k=1..n) - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Feb 05 2005

a(n)=sum{k=0..n, S2(n+1, k+1)k!} - Paul Barry (pbarry(AT)wit.ie), Apr 20 2005

A000629 = binomial transform of this sequence. a(n) = sum of terms in n-th row of A028246 - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 30 2005

a(n) = 2*(-1)^n * n!*Laguerre(n,P((.),2)), umbrally, where P(j,t) are the polynomials in A131758. - Tom Copeland (tcjpn(AT)msn.com), Sep 28 2007

a(3)=6: the necklace representatives on 1,2,3 are ({123}), ({12},{3}), ({13},{2}), ({23},{1}), ({1},{2},{3}), ({1},{3},{2})

spec := [ B, {B=Cycle(Set(Z, card>=1))}, labeled ]; [seq(combstruct[count](spec, size=n), n=0..20)];

a:=n->add(stirling2(n, k)*(k-1)!, k=1..n); (Zabrocki)

a[ 0 ] = 1; a[ n_ ] := (a[ n ] = 1+Sum[ Binomial[ n, k ] a[ n-k ], {k, 1, n} ])

(PARI) a(n)=if(n<0, 0, n!*polcoeff(subst((1+y)/(1-y), y, exp(x+x*O(x^n))-1), n))

Same as A076726 except for a(0). Cf. A008965.

Binomial transform of A000670, also double of A000670 - Joe Keane (jgk(AT)jgk.org)

A002050(n) = a(n) - 1.

Cf. A008277.

Adjacent sequences: A000626 A000627 A000628 this_sequence A000630 A000631 A000632

Sequence in context: A052844 A052859 A103937 this_sequence A032187 A003659 A032271

nonn,easy,eigen,nice

njas, D. E. Knuth, Nick Singer (nsinger(AT)eos.hitc.com)