|
| |
|
|
A000587
|
|
Rao Uppuluri-Carpenter numbers (or complementary Bell numbers): e.g.f. = exp(1 - exp(x)).
(Formerly M1913 N0755)
|
|
35
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,6
|
|
|
COMMENTS
|
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
Row sums of triangle A144185 = a signed, shifted version of A000587: (1, 0, 1, 1, 2, 9, -9, 50, -267, -413, -2180,...). Triangle A144185 is generated from A118433, the self-inverse triangle. [From Gary W. Adamson, Sep 13 2008]
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. [From 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
|
|
|
REFERENCES
|
R. E. Beard, On the Coefficients in the Expansion of e^e^t and e^-e^t, J. Institute of Actuaries, 76 (1950), 152-163.
B. Harris and L. Schoenfeld, Asymptotic expansions for the coefficients of analytic functions, Ill. J. Math., 12 (1968), 264-277.
M. Klazar, Counting even and odd partitions, Amer. Math. Monthly, 110 (No. 6, 2003), 527-532.
M. Klazar, Bell numbers, their relatives and algebraic differential equations, J. Combin. Theory, A 102 (2003), 63-87.
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.
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.
V. R. Rao Uppuluri and J. A. Carpenter, Numbers generated by the function exp(1-e^x), Fib. Quart. 7 (1969), 437-448.
Alfréd Rényi, Új modszerek es eredmenyek a kombinatorikus analfzisben. I. MTA III Oszt. Ivozl., 16 (1966), 7-105.
Frank Ruskey and Jennifer Woodcock, The Rand and block distances of pairs of set partitions, in International Workshop on Combinatorial Algorithms, Victoria, 2011. LNCS, to appear.
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. - From N. J. A. Sloane, Oct 03 2012
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).
Subbarao, M. V. and Verma, A., Some remarks on a product expansion. An unexplored partition function, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (Gainesville, FL, 1999), 267-283, Kluwer, Dordrecht, 2001.
A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, Arxiv preprint arXiv:1107.2938, 2011.
Yang, Yifan, On a multiplicative partition function, Electron. J. Combin. 8 (2001), no. 1, Research Paper 19.
|
|
|
LINKS
|
T. D. Noe, Table of n, a(n) for n=0..100
Tewodros Amdeberhan, Valerio de Angelis and Victor H. Moll, Complementary Bell numbers: arithmetical properties and Wilf's conjecture
S. de Wannemacker, T. Laffey and R. Osburn, On a conjecture of Wilf
S. Ramanujan, Notebook entry
Eric Weisstein's World of Mathematics, Complementary Bell Number
|
|
|
FORMULA
|
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 to n) (-1)^k S(n, k) ), where S(i, j) are 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/prod[l=1..k, 1+lx]}.
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*A000587 with general term Sum( (k=0 to 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
G.f.: -1/U(0) where U(k)= x*k - 1 - x + x^2*(k+1)/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Sep 28 2012
G.f.: 1/(U(0)+x) where U(k)= 1 + x - x*(k+1)/(1 + x/U(k+1)) ; (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 12 2012
G.f.: 1+x/G(0) where G(k)= x*k - 1 + x^2*(k+1)/G(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Nov 04 2012
G.f.: (1 - G(0) )/(x+1) where G(k) = 1 - 1/(1-k*x)/(1-x/(x+1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 17 2013
G.f.: 1 + x/(G(0)-x) where G(k) = x*k +2*x - 1 - x*(x*k+x-1)/G(k+1) ; (recursively defined continued fraction ). - Sergei N. Gladkovskii, Jan 28 2013
|
|
|
EXAMPLE
|
1 - x + x^3 + x^4 - 2*x^5 - 9*x^6 - 9*x^7 + 50*x^8 + 267*x^9 + 413*x^10 - ...
|
|
|
MAPLE
|
# Uses floating point arithmetic, increase working precision for n>60.
A000587 := proc(n) local a, b, i;
if n = 0 then 1 else a := [seq(2, i=1..n-1)]; b := [seq(1, i=1..n-1)];
-exp(-x)*hypergeom(a, b, x); round(evalf(subs(x=-1, %), 66)) fi end:
seq(A000587(n), n=0..25); # - Peter Luschny, Mar 30 2011
|
|
|
MATHEMATICA
|
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]!] (* From Harvey P. Dale, Nov 04 2011 *)
|
|
|
PROG
|
(Sage) expnums(26, -1)# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009]
(PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( 1 - exp( x + x * O(x^n))), n))} /* Michael Somos, Mar 14 2011 */
(PARI) {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 */
(PARI) Vec(serlaplace(exp(1 - exp(x+O(x^99))))) /* Joerg Arndt, Apr 1 2011 */
(Python) 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(0, 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
|
|
|
CROSSREFS
|
Cf. A000110, A011971 (base triangle PE), A078937 = PE^2
Cf. A144185, A118433 [From Gary W. Adamson, Sep 13 2008]
Sequence in context: A198942 A168333 A109322 * A014182 A131463 A065644
Adjacent sequences: A000584 A000585 A000586 * A000588 A000589 A000590
|
|
|
KEYWORD
|
sign,easy,nice,changed
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
STATUS
|
approved
|
| |
|
|
