login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A001861 Expansion of e.g.f. exp(2*(exp(x) - 1)).
(Formerly M1662 N0653)
44
1, 2, 6, 22, 94, 454, 2430, 14214, 89918, 610182, 4412798, 33827974, 273646526, 2326980998, 20732504062, 192982729350, 1871953992254, 18880288847750, 197601208474238, 2142184050841734, 24016181943732414, 278028611833689478 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Values of Bell polynomials: ways of placing n labeled balls into n unlabeled (but 2-colored) boxes.

First column of the square of the matrix exp(P)/exp(1) given in A011971. - Gottfried Helms, Mar 30 2007

Base matrix in A011971, second power in A078937, third power in A078938, fourth power in A078939. - Gottfried Helms, Apr 08 2007

Equals row sums of triangle A144061. - Gary W. Adamson, Sep 09 2008

Equals eigensequence of triangle A109128. - Gary W. Adamson, Apr 17 2009

Hankel transform is A108400. - Paul Barry, Apr 29 2009

The number of ways of putting n labeled balls into a set of bags and then putting the bags into 2 labeled boxes. An example is given below. - Peter Bala, Mar 23 2013

The f-vectors of n-dimensional hypercube are given by A038207 = exp[M*B(.,2)] = exp[M*A001861(.)] where M = A238385-I and (B(.,x))^n = B(n,x) are the Bell polynomials (cf. A008277). - Tom Copeland, Apr 17 2014

Moments of the Poisson distribution with mean 2. - Vladimir Reshetnikov, May 17 2016

Exponential self-convolution of Bell numbers (A000110). - Vladimir Reshetnikov, Oct 06 2016

REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

LINKS

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

M. Aigner, A characterization of the Bell numbers, Discr. Math., 205 (1999), 207-210.

C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, Generating Functions for Generating Trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.

Jacques Carlier and Corinne Lucet, A decomposition algorithm for network reliability evaluation. In First International Colloquium on Graphs and Optimization (GOI), 1992 (Grimentz). Discrete Appl. Math. 65 (1996), 141-156 (see page 152 and Fig 6).

Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012. - From N. J. A. Sloane, Sep 17 2012

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 66

G. Labelle et al., Stirling numbers interpolation using permutations with forbidden subsequences, Discrete Math. 246 (2002), 177-195.

T. Mansour, M. Shattuck and D. G. L. Wang, Recurrence relations for patterns of type (2, 1) in flattened permutations, arXiv preprint arXiv:1306.3355 [math.CO], 2013

T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]

OEIS Wiki, Sorting numbers

Frank Simon, Algebraic Methods for Computing the Reliability of Networks, Dissertation, Doctor Rerum Naturalium (Dr. rer. nat.), Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2012. See Table 5.1. - From N. J. A. Sloane, Jan 04 2013

Amit Kumar Singh, Akash Kumar and Thambipillai Srikanthan, Accelerating Throughput-aware Run-time Mapping for Heterogeneous MPSoCs, ACM Transactions on Design Automation of Electronic Systems, 2012. - From N. J. A. Sloane, Dec 24 2012

FORMULA

a(n) = Sum_{k=1..n} 2^k*stirling2(n, k). - Emeric Deutsch, Oct 20 2001

a(n) = exp(-2)*Sum_{k>=1} 2^k*k^n/k!. - Benoit Cloitre, Sep 25 2003

G.f. satisfies 2*(x/(1-x))*A(x/(1-x)) = A(x) - 1; twice the binomial transform equals the sequence shifted one place left. - Paul D. Hanna, Dec 08 2003

PE = exp(matpascal(5)-matid(6)); A = PE^2; a(n)=A[n,1]. - Gottfried Helms, Apr 08 2007

G.f.: 1/(1-2x-2x^2/(1-3x-4x^2/(1-4x-6x^2/(1-5x-8x^2/(1-6x-10x^2/(1-... (continued fraction). - Paul Barry, Apr 29 2009

O.g.f.: Sum_{n>=0} 2^n*x^n / Product_{k=1..n} (1-k*x). - Paul D. Hanna, Feb 15 2012

a(n) ~ exp(-2-n+n/LambertW(n/2))*n^n/LambertW(n/2)^(n+1/2). - Vaclav Kotesovec, Jan 06 2013

G.f.: (G(0) - 1)/(x-1)/2 where G(k) =  1 - 2/(1-k*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2013

G.f.: 1/Q(0) where Q(k) =  1 + x*k - x - x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013

G.f.: ((1+x)/Q(0)-1)/(2*x), where Q(k)= 1 - (k+1)*x - 2*(k+1)*x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013

G.f.: T(0)/(1-2*x), where T(k) = 1 - 2*x^2*(k+1)/( 2*x^2*(k+1) - (1-2*x-x*k)*(1-3*x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 24 2013

a(n) = Sum_{k=0..n} A033306(n,k) = Sum_{k=0..n} binomial(n,k)*Bell(k)*Bell(n-k), where Bell = A000110 (see Motzkin, p. 170). - Danny Rorabaugh, Oct 18 2015

EXAMPLE

a(2) = 6: The six ways of putting 2 balls into bags (denoted by { }) and then into 2 labeled boxes (denoted by [ ]) are

01: [{1,2}] [ ];

02: [ ] [{1,2}];

03: [{1}] [{2}];

04: [{2}] [{1}];

05: [{1} {2}] [ ];

06: [ ] [{1} {2}].

- Peter Bala, Mar 23 2013

MAPLE

with(combinat); A001861:=n->add(stirling2(n, k)*2^k, k=0..n); seq(A001861(n), n=0..20); # Wesley Ivan Hurt, Apr 18 2014

MATHEMATICA

Table[Sum[StirlingS2[n, k]*2^k, {k, 0, n}], {n, 0, 21}] (* Geoffrey Critzer, Oct 06 2009 *)

mx = 16; p = 1; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)

Table[BellB[n, 2], {n, 0, 20}] (* Vaclav Kotesovec, Jan 06 2013 *)

PROG

(PARI) a(n)=if(n<0, 0, n!*polcoeff(exp(2*(exp(x+x*O(x^n))-1)), n))

(PARI) {a(n)=polcoeff(sum(m=0, n, 2^m*x^m/prod(k=1, m, 1-k*x +x*O(x^n))), n)} /* Paul D. Hanna, Feb 15 2012 */

(Sage) expnums(30, 2) # Zerinvary Lajos, Jun 26 2008

CROSSREFS

For boxes of 1 color, see A000110, for 3 colors see A027710, for 4 colors see A078944, for 5 colors see A144180, for 6 colors see A144223, for 7 colors see A144263, for 8 colors see A221159.

First column of A078937.

Equals 2*A035009(n), n>0.

Row sums of A033306, A036073, A049020, and A144061.

Cf. A000110, A000587, A002871, A027710, A056857, A068199, A068200, A068201, A078937, A078938, A078944, A078945, A109128, A129323, A129324, A129325, A129327, A129328, A129329, A129331, A129332, A129333, A144180, A144223, A144263, A189233, A221159, A221176.

Sequence in context: A109317 A109153 A030453 * A049526 A187251 A193763

Adjacent sequences:  A001858 A001859 A001860 * A001862 A001863 A001864

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Simon Plouffe

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 25 12:08 EDT 2017. Contains 287027 sequences.