A078937 -id:A078937 - cp's OEIS frontend

A000587 Rao Uppuluri-Carpenter numbers (or complementary Bell numbers): e.g.f. = exp(1 - exp(x)).

1, -1, 0, 1, 1, -2, -9, -9, 50, 267, 413, -2180, -17731, -50533, 110176, 1966797, 9938669, 8638718, -278475061, -2540956509, -9816860358, 27172288399, 725503033401, 5592543175252, 15823587507881, -168392610536153, -2848115497132448, -20819319685262839

Offset: 0

N. J. A. Sloane

Alternating row sums of Stirling2 triangle A048993.
Related to the matrix-exponential of the Pascal-matrix, see A000110 and A011971. - Gottfried Helms, Apr 08 2007
Closely linked to A000110 and especially the contribution there of Jonathan R. Love (japanada11(AT)yahoo.ca), Feb 22 2007, by offering what is a complementary finding.
Number of set partitions of 1..n with an even number of parts, minus the number of such partitions with an odd number of parts. - Franklin T. Adams-Watters, May 04 2010
After -2, the smallest prime is a(36) = -1454252568471818731501051, no others through a(100).  What is the first prime >0 in the sequence? - Jonathan Vos Post, Feb 02 2011
a(723) ~ 1.9*10^1265 is almost certainly prime. - D. S. McNeil, Feb 02 2011
Stirling transform of a(n) = [1, -1, 0, 1, 1, ...] is A033999(n) = [1, -1, 1, -1, 1, ...]. - Michael Somos, Mar 28 2012
Negated coefficients in the asymptotic expansion: A005165(n)/n! ~ 1 - 1/n + 1/n^2 + 0/n^3 - 1/n^4 - 1/n^5 + 2/n^6 + 9/n^7 + 9/n^8 - 50/n^9 - 267/n^10 - 413/n^11 + O(1/n^12), starting from the O(1/n) term. - Vladimir Reshetnikov, Nov 09 2016
Named after Venkata Ramamohana Rao Uppuluri and John A. Carpenter of the Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee. They are called "Rényi numbers" by Fekete (1999), after the Hungarian mathematician Alfréd Rényi (1921-1970). - Amiram Eldar, Mar 11 2022

			G.f. = 1 - x + x^3 + x^4 - 2*x^5 - 9*x^6 - 9*x^7 + 50*x^8 + 267*x^9 + 413*x^10 - ...

N. A. Kolokolnikova, Relations between sums of certain special numbers (Russian), in Asymptotic and enumeration problems of combinatorial analysis, pp. 117-124, Krasnojarsk. Gos. Univ., Krasnoyarsk, 1976.
Alfréd Rényi, Új modszerek es eredmenyek a kombinatorikus analfzisben. I. MTA III Oszt. Ivozl., Vol. 16 (1966), pp. 7-105.
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).
M. V. Subbarao and A. Verma, Some remarks on a product expansion. An unexplored partition function, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (Gainesville, FL, 1999), pp. 267-283, Kluwer, Dordrecht, 2001.

Alois P. Heinz, Table of n, a(n) for n = 0..595 (first 101 terms from T. D. Noe)
M. Aguiar and A. Lauve, The characteristic polynomial of the Adams operators on graded connected Hopf algebras, 2014. See Example 31. - _N. J. A. Sloane_, May 24 2014
W. Asakly, A. Blecher, C. Brennan, A. Knopfmacher, T. Mansour, and S. Wagner, Set partition asymptotics and a conjecture of Gould and Quaintance, Journal of Mathematical Analysis and Applications, Volume 416, Issue 2 (15 August 2014), Pages 672-682.
Tewodros Amdeberhan, Valerio de Angelis and Victor H. Moll, Complementary Bell numbers: arithmetical properties and Wilf's conjecture.
S. Barbero, U. Cerruti, and N. Murru, A Generalization of the Binomial Interpolated Operator and its Action on Linear Recurrent Sequences, J. Int. Seq., Vol. 13 (2010), Article 10.9.7.
R. E. Beard, On the Coefficients in the Expansion of e^(e^t) and e^(-e^t), J. Institute of Actuaries, Vol. 76 (1950), pp. 152-163. [Annotated scanned copy]
Pascal Caron, Jean-Gabriel Luque, Ludovic Mignot, and Bruno Patrou, State complexity of catenation combined with a boolean operation: a unified approach, arXiv:1505.03474 [cs.FL], 2015.
Valerio De Angelis and Dominic Marcello, Wilf's Conjecture, The American Mathematical Monthly, Vol. 123, No. 6 (2016), pp. 557-573.
S. de Wannemacker, T. Laffey and R. Osburn, On a conjecture of Wilf, arXiv:math/0608085 [math.NT], 2006-2007.
Branko Dragovich, On Summation of p-Adic Series, arXiv:1702.02569 [math.NT], 2017.
Branko Dragovich, Andrei Yu. Khrennikov, and Natasa Z. Misic, Summation of p-Adic Functional Series in Integer Points, arXiv:1508.05079, 2015
B. Dragovich and N. Z. Misic, p-Adic invariant summation of some p-adic functional series, P-Adic Numbers, Ultrametric Analysis, and Applications, Volume 6, Issue 4 (October 2014), pp. 275-283.
Antal E. Fekete, Apropos Bell and Stirling Numbers, Crux Mathematicorum, Vol. 25, No. 5 (1999), pp. 274-281.
B. Harris and L. Schoenfeld, Asymptotic expansions for the coefficients of analytic functions, Ill. J. Math., Vol. 12 (1968), pp. 264-277.
M. Klazar, Counting even and odd partitions, Amer. Math. Monthly, Vol. 110, No. 6 (2003), pp. 527-532.
M. Klazar, Bell numbers, their relatives and algebraic differential equations, J. Combin. Theory, A 102 (2003), 63-87.
A. Knopfmacher and M. E. Mays, Graph compositions I: Basic enumerations, Integers, Vol. 1 (2001), Article A4. (See the first two columns of the table on p. 9.)
Vaclav Kotesovec, Plot of |a(n)/n!|^(1/n) / |exp(1/W(-n))/W(-n)| for n = 1..40000, where W is the LambertW function.
Peter J. Larcombe, Jack Sutton, and James Stanton, A note on the constant 1/e, Palest. J. Math. (2023) Vol. 12, No. 2, 609-619. See p. 617.
J. W. Layman and C. L. Prather, Generalized Bell numbers and zeros of successive derivatives of an entire function, Journal of Mathematical Analysis and Applications, Volume 96, Issue 1 (15 October 1983), Pages 42-51.
Toufik Mansour and Mark Shattuck, Counting subword patterns in permutations arising as flattened partitions of sets, Appl. Anal. Disc. Math. (2022), OnLine-First (00):9-9.
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.
S. Ramanujan, Notebook entry.
V. R. Rao Uppuluri and J. A. Carpenter, Numbers generated by the function exp(1-e^x), Fib. Quart., Vol. 7, No. 4 (1969), pp. 437-448.
Frank Ruskey and Jennifer Woodcock, The Rand and block distances of pairs of set partitions, Combinatorial algorithms, 287-299, Lecture Notes in Comput. Sci., 7056, Springer, Heidelberg, 2011.
Frank Ruskey, Jennifer Woodcock and Yuji Yamauchi, Counting and computing the Rand and block distances of pairs of set partitions, Journal of Discrete Algorithms, Volume 16 (October 2012), Pages 236-248. [_N. J. A. Sloane_, Oct 03 2012]
M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq., Vol. 14 (2011), Article 11.9.7.
D. Subedi, Complementary Bell Numbers and p-adic Series, J. Int. Seq., Vol. 17 (2014), Article 14.3.1.
A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, arXiv preprint arXiv:1107.2938 [math.NT], 2011.
Eric Weisstein's World of Mathematics, Complementary Bell Number.
D. Wuilquin, Letters to N. J. A. Sloane, August 1984.
Yifan Yang, On a multiplicative partition function, Electron. J. Combin., Vol. 8, No. 1 (2001), Research Paper 19.

Cf. A000110, A011971 (base triangle PE), A078937 (PE^2).

Cf. A007318, A153229, A213170.

a000587 n = a000587_list !! n
a000587_list = 1 : f a007318_tabl [1] where
   f (bs:bss) xs = y : f bss (y : xs) where y = - sum (zipWith (*) xs bs)
-- Reinhard Zumkeller, Mar 04 2014

b:= proc(n, t) option remember; `if`(n=0, 1-2*t,
      add(b(n-j, 1-t)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 28 2016

Table[ -1 * Sum[ (-1)^( k + 1) StirlingS2[ n, k ], {k, 0, n} ], {n, 0, 40} ]
With[{nn=30},CoefficientList[Series[Exp[1-Exp[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 04 2011 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ 1 - Exp[x]], {x, 0, n}]]; (* Michael Somos, May 27 2014 *)
a[ n_] := If[ n < 0, 0, With[{m = n + 1}, SeriesCoefficient[ Series[ Nest[ x Factor[ 1 - # /. x -> x / (1 - x)] &, 0, m], {x, 0, m}], {x, 0, m}]]]; (* Michael Somos, May 27 2014 *)
Table[BellB[n, -1], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)
b[1] = 1; k = 1; Flatten[{1, Table[Do[j = k; k -= b[m]; b[m] = j;, {m, 1, n-1}]; b[n] = k; k*(-1)^n, {n, 1, 40}]}] (* Vaclav Kotesovec, Sep 09 2019 *)

{a(n) = if( n<0, 0, n! * polcoeff( exp( 1 - exp( x + x * O(x^n))), n))}; /* Michael Somos, Mar 14 2011 */

{a(n) = local(A); if( n<0, 0, n++; A = O(x); for( k=1, n, A = x - x * subst(A, x, x / (1 - x))); polcoeff( A, n))}; /* Michael Somos, Mar 14 2011 */

Vec(serlaplace(exp(1 - exp(x+O(x^99))))) /* Joerg Arndt, Apr 01 2011 */

a(n)=round(exp(1)*suminf(k=0,(-1)^k*k^n/k!))
vector(20,n,a(n-1)) \\ Derek Orr, Sep 19 2014 -- a direct approach

x='x+O('x^66); Vec(serlaplace(exp(1 - exp(x)))) \\ Michel Marcus, Sep 19 2014

# The objective of this implementation is efficiency.
# n -> [a(0), a(1), ..., a(n)] for n > 0.
def A000587_list(n):
    A = [0 for i in range(n)]
    A[n-1] = 1
    R = [1]
    for j in range(0, n):
        A[n-1-j] = -A[n-1]
        for k in range(n-j, n):
            A[k] += A[k-1]
        R.append(A[n-1])
    return R
# Peter Luschny, Apr 18 2011

# Python 3.2 or higher required
from itertools import accumulate
A000587, blist, b = [1,-1], [1], -1
for _ in range(30):
    blist = list(accumulate([b]+blist))
    b = -blist[-1]
    A000587.append(b) # Chai Wah Wu, Sep 19 2014

expnums(26, -1) # Zerinvary Lajos, May 15 2009

a(n) = e*Sum_{k>=0} (-1)^k*k^n/k!. - Benoit Cloitre, Jan 28 2003
E.g.f.: exp(1 - e^x).
a(n) = Sum_{k=0..n} (-1)^k S2(n, k), where S2(i, j) are the Stirling numbers of second kind A008277.
G.f.: (x/(1-x))*A(x/(1-x)) = 1 - A(x); the binomial transform equals the negative of the sequence shifted one place left. - Paul D. Hanna, Dec 08 2003
With different signs: g.f.: Sum_{k>=0} x^k/Product_{L=1..k} (1 + L*x).
Recurrence: a(n) = -Sum_{i=0..n-1} a(i)*C(n-1, i). - Ralf Stephan, Feb 24 2005
Let P be the lower-triangular Pascal-matrix, PE = exp(P-I) a matrix-exponential in exact integer arithmetic (or PE = lim exp(P)/exp(1) as limit of the exponential); then a(n) = PE^-1 [n,1]. - Gottfried Helms, Apr 08 2007
Take the series 0^n/0! - 1^n/1! + 2^n/2! - 3^n/3! + 4^n/4! + ... If n=0 then the result will be 1/e, where e = 2.718281828... If n=1, the result will be -1/e. If n=2, the result will be 0 (i.e., 0/e). As we continue for higher natural number values of n sequence for the Roa Uppuluri-Carpenter numbers is generated in the numerator, i.e., 1/e, -1/e, 0/e, 1/e, 1/e, -2/e, -9/e, -9/e, 50/e, 267/e, ... . - Peter Collins (pcolins(AT)eircom.net), Jun 04 2007
The sequence (-1)^n*a(n), with general term Sum_{k=0..n} (-1)^(n-k)*S2(n, k), has e.g.f. exp(1-exp(-x)). It also has Hankel transform (-1)^C(n+1,2)*A000178(n) and binomial transform A109747. - Paul Barry, Mar 31 2008
G.f.: 1 / (1 + x / (1 - x / (1 + x / (1 - 2*x / (1 + x / (1 - 3*x / (1 + x / ...))))))). - Michael Somos, May 12 2012
From Sergei N. Gladkovskii, Sep 28 2012 to Feb 07 2014: (Start)
Continued fractions:
G.f.: -1/U(0) where U(k) = x*k - 1 - x + x^2*(k+1)/U(k+1).
G.f.: 1/(U(0)+x) where U(k) = 1 + x - x*(k+1)/(1 + x/U(k+1)).
G.f.: 1+x/G(0) where G(k) = x*k - 1 + x^2*(k+1)/G(k+1).
G.f.: (1 - G(0))/(x+1) where G(k) = 1 - 1/(1-k*x)/(1-x/(x+1/G(k+1) )).
G.f.: 1 + x/(G(0)-x) where G(k) = x*k + 2*x - 1 - x*(x*k+x-1)/G(k+1).
G.f.: G(0)/(1+x), where G(k) = 1-x^2*(k+1)/(x^2*(k+1)+(x*k-1-x)*(x*k-1)/G(k+1)).
(End)
a(n) = B_n(-1), where B_n(x) is n-th Bell polynomial. - Vladimir Reshetnikov, Oct 20 2015
From Mélika Tebni, May 20 2022: (Start)
a(n) = Sum_{k=0..n} (-1)^k*Bell(k)*A129062(n, k).
a(n) = Sum_{k=0..n} (-1)^k*k!*A130191(n, k). (End)

A001861 Expansion of e.g.f. exp(2*(exp(x) - 1)).

Original entry on oeis.org

1, 2, 6, 22, 94, 454, 2430, 14214, 89918, 610182, 4412798, 33827974, 273646526, 2326980998, 20732504062, 192982729350, 1871953992254, 18880288847750, 197601208474238, 2142184050841734, 24016181943732414, 278028611833689478, 3319156078802044158, 40811417293301014150

Offset: 0

N. J. A. Sloane, Simon Plouffe

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

			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

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).

Seiichi Manyama, Table of n, a(n) for n = 0..558 (terms 0..100 from T. D. Noe)
M. Aigner, A characterization of the Bell numbers, Discr. Math., 205 (1999), 207-210.
Michael Anshelevich, Product formulas on posets, Wick products, and a correction for the q-Poisson process, arXiv:1708.08034 [math.OA], 2017, See Proposition 34 p. 25.
Diego Arcis, Camilo González, and Sebastián Márquez, Symmetric functions in noncommuting variables in superspace, arXiv:2312.00574 [math.CO], 2023.
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.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
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
Wan-Ming Guo and Lily Li Liu, Asymptotic normality of the Stirling-Whitney-Riordan triangle, Filomat (2023) Vol. 37, No. 9, 2923-2934.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 66 [broken link?]
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
G. Labelle et al., Stirling numbers interpolation using permutations with forbidden subsequences, Discrete Math. 246 (2002), 177-195.
Huyile Liang, Jeffrey Remmel, and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see page 20.
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.
Victor Meally, Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.
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
J. Riordan, Letter to N. J. A. Sloane, Oct. 1970
J. Riordan, Letter, Oct 31 1977
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
Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.

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, A213170, A221159, A221176.

[&+[2^k*StirlingSecond(n, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, May 18 2019

A001861:=n->add(Stirling2(n,k)*2^k, k=0..n); seq(A001861(n), n=0..20); # Wesley Ivan Hurt, Apr 18 2014
# second Maple program:
b:= proc(n, m) option remember;
     `if`(n=0, 2^m, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 04 2021

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 *)

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

{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 */

{a(n) = sum(k=0, n, 2^k*stirling(n, k, 2))} \\ Seiichi Manyama, Jul 28 2019

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

a(n) = Sum_{k=0..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
a(0) = 1 and a(n) = 2 * Sum_{k=0..n-1} binomial(n-1,k)*a(k) for n > 0. - Seiichi Manyama, Sep 25 2017 [corrected by Ilya Gutkovskiy, Jul 12 2020]

A027710 Number of ways of placing n labeled balls into n unlabeled (but 3-colored) boxes.

Original entry on oeis.org

1, 3, 12, 57, 309, 1866, 12351, 88563, 681870, 5597643, 48718569, 447428856, 4318854429, 43666895343, 461101962108, 5072054649573, 57986312752497, 687610920335610, 8442056059773267, 107135148331162767, 1403300026585387686, 18946012544520590991

Offset: 0

George Yuhasz (gyuhasz(AT)vt.edu) and John W. Layman

nonn

Binomial transform of this sequence is A078940 and a(n+1) = 3*A078940(n). - Paul D. Hanna, Dec 08 2003
First column of the cube 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.
The number of ways of putting n labeled balls into a set of bags and then putting the bags into 3 labeled boxes. - Peter Bala, Mar 23 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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
Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.

Cf. A000110, A001861, A056857, A078937, A078938, A078940, A078944, A078945, A129323, A129324, A129325, A129327, A129328, A129329, A129331, A129332, A129333, A144180, A144223, A144263, A189233, A221159, A221176.

b:= proc(n, m) option remember; `if`(n=0,
      1, m*b(n-1, m)+3*b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..27);  # Alois P. Heinz, Aug 03 2021

colors=3; Array[ bell, 25 ]; For[ x=1, x<=25, x++, bell[ x ]=0 ]; bell[ 1 ]=colors;
Print[ "1 ", colors ]; For[ n=2, n<=25, n++, bell[ n ]=colors*bell[ n-1 ];
For[ i=1, n-i>1, i++, bell[ n-i ]=bell[ n-i ]*(n-i)+colors*bell[ n-i-1 ] ];
bellsum=0; For[ t=0, tVaclav Kotesovec, Mar 12 2014 *)

a(n)=if(n<0,0,n!*polcoeff(exp(3*(exp(x+x*O(x^n))-1)),n))

from sage.combinat.expnums import expnums2
expnums(22, 3) # Zerinvary Lajos, Jun 26 2008

E.g.f.: exp {3(e^x-1)}. - Michael Somos, Oct 18 2002
a(n) = exp(-3)*Sum_{k>=0} 3^k*k^n/k!. - Benoit Cloitre, Sep 25 2003
G.f.: 3*(x/(1-x))*A(x/(1-x)) = A(x) - 1; thrice the binomial transform equals the sequence shifted one place left. - Paul D. Hanna, Dec 08 2003
a(n) = Sum_{k = 0..n} 3^k*A048993(n, k); A048993: Stirling2 numbers. - Philippe Deléham, May 09 2004
PE=exp(matpascal(5))/exp(1); A = PE^3; a(n)= A[ n,1 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^3; a(n)=A[ n,1]. - Gottfried Helms, Apr 08 2007
G.f.: (G(0) - 1)/(x-1)/3 where G(k) =  1 - 3/(1-k*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
G.f.: T(0)/(1-3*x), where T(k) = 1 - 3*x^2*(k+1)/( 3*x^2*(k+1) - (1-3*x-x*k)*(1-4*x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 24 2013
a(n) ~ n^n * exp(n/LambertW(n/3)-3-n) / (sqrt(1+LambertW(n/3)) * LambertW(n/3)^n). - Vaclav Kotesovec, Mar 12 2014
G.f.: Sum_{j>=0} 3^j*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 07 2019

Entry revised by N. J. A. Sloane, Apr 25 2007

A078944 First column of A078939, the fourth power of lower triangular matrix A056857.

Original entry on oeis.org

1, 4, 20, 116, 756, 5428, 42356, 355636, 3188340, 30333492, 304716148, 3218555700, 35618229364, 411717043252, 4957730174836, 62045057731892, 805323357485684, 10820999695801908, 150271018666120564, 2153476417340487476

Offset: 0

Paul D. Hanna, Dec 18 2002

nonn

Also, the number of ways of placing n labeled balls into n unlabeled (but 4-colored) boxes. Binomial transform of this sequence is A078945 and a(n+1) = 4*A078945(n). - Paul D. Hanna, Dec 08 2003
First column of PE^4, where PE is given in A011971, second power in A078937, third power in A078938, fourth power in A078939. - Gottfried Helms, Apr 08 2007
The number of ways of putting n labeled balls into a set of bags and then putting the bags into 4 labeled boxes. - Peter Bala, Mar 23 2013
Exponential self-convolution of A001861. - Vladimir Reshetnikov, Oct 06 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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

Cf. A000110, A001861, A027710, A056857, A078937, A078938, A078939, A078944, A078945, A129323, A129324, A129325, A129327, A129328, A129329, A129331, A129332, A129333, A144180, A144223, A144263, A189233, A221159, A221176.

A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A078938 := proc(n,c) add( A078937(n,k)*A056857(k+1,c),k=0..n) ; end: A078939 := proc(n,c) add( A078938(n,k)*A056857(k+1,c),k=0..n) ; end: A078944 := proc(n) A078939(n+1,0) ; end: seq(A078944(n),n=0..25) ; # R. J. Mathar, May 30 2008
# second Maple program:
b:= proc(n, m) option remember; `if`(n=0, 4^m,
      add(b(n-1, max(m, j)), j=1..m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 03 2021

Table[n!, {n, 0, 20}]CoefficientList[Series[E^(4E^x-4), {x, 0, 20}], x]
Table[BellB[n,4],{n,0,20}] (* Vaclav Kotesovec, Mar 12 2014 *)
With[{nn=20},CoefficientList[Series[Exp[4(Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 03 2022 *)

expnums(20, 4) # Zerinvary Lajos, Jun 26 2008

PE=exp(matpascal(5))/exp(1); A = PE^4; a(n)= A[ n,1 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^4; a(n)=A[ n,1]. - Gottfried Helms, Apr 08 2007
E.g.f.: exp(4*(exp(x)-1)).
a(n) = exp(-4)*Sum_{k>=0} 4^k*k^n/k!. - Benoit Cloitre, Sep 25 2003
G.f.: 4*(x/(1-x))*A(x/(1-x)) = A(x) - 1; four times the binomial transform equals this sequence shifted one place left. - Paul D. Hanna, Dec 08 2003
a(n) = Sum_{k = 0..n} 4^k*A048993(n, k); A048993: Stirling2 numbers. - Philippe Deléham, May 09 2004
G.f.: (G(0) - 1)/(x-1)/4 where G(k) = 1 - 4/(1-k*x)/(1-x/(x-1/G(k+1))); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
G.f.: T(0)/(1-4*x), where T(k) = 1 - 4*x^2*(k+1)/(4*x^2*(k+1) - (1-(k+4)*x)*(1-(k+5)*x)/T(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 28 2013
a(n) ~ n^n * exp(n/LambertW(n/4)-4-n) / (sqrt(1+LambertW(n/4)) * LambertW(n/4)^n). - Vaclav Kotesovec, Mar 12 2014
G.f.: Sum_{j>=0} 4^j*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 07 2019

More terms from R. J. Mathar, May 30 2008

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar

cp's OEIS Frontend

A000587 Rao Uppuluri-Carpenter numbers (or complementary Bell numbers): e.g.f. = exp(1 - exp(x)).

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A001861 Expansion of e.g.f. exp(2*(exp(x) - 1)).

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A027710 Number of ways of placing n labeled balls into n unlabeled (but 3-colored) boxes.

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A078944 First column of A078939, the fourth power of lower triangular matrix A056857.

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A078938 Cube of lower triangular matrix of A056857 (successive equalities in set partitions of n).

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A078939 Fourth power of lower triangular matrix of A056857 (successive equalities in set partitions of n).