%I A000587 M1913 N0755
%S A000587 1,1,0,1,1,2,9,9,50,267,413,2180,17731,50533,110176,1966797,
%T A000587 9938669,8638718,278475061,2540956509,9816860358,27172288399,
%U A000587 725503033401,5592543175252,15823587507881,168392610536153
%V A000587 1,-1,0,1,1,-2,-9,-9,50,267,413,-2180,-17731,-50533,110176,1966797,
%W A000587 9938669,8638718,-278475061,-2540956509,-9816860358,27172288399,
%X A000587 725503033401,5592543175252,15823587507881,-168392610536153
%N A000587 Rao Uppuluri-Carpenter numbers: e.g.f. = exp (1 - e^x).
%C A000587 Alternating row sums of Stirling2 triangle A048993.
%C A000587 Related to the matrix-exponential of the Pascal-matrix, see A000110 and
A011971. - Gottfried Helms (helms(AT)uni-kassel.de) Apr 08 2007
%C A000587 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 (qntmpkt(AT)yahoo.com), Sep 13 2008]
%D A000587 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000587 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000587 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.
%D A000587 B. Harris and L. Schoenfeld, Asymptotic expansions for the coefficients
of analytic functions, Ill. J. Math., 12 (1968), 264-277.
%D A000587 M. Klazar, Counting even and odd partitions, Amer. Math. Monthly, 110
(No. 6, 2003), 527-532.
%D A000587 M. Klazar, Bell numbers, their relatives and algebraic differential equations,
J. Combin. Theory, A 102 (2003), 63-87.
%D A000587 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.
%D A000587 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.
%D A000587 V. R. Rao Uppuluri and J. A. Carpenter, Numbers generated by the function
exp(1-e^x), Fib. Quart. 7 (1969), 437-448.
%D A000587 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.
%D A000587 Yang, Yifan, On a multiplicative partition function, Electron. J. Combin.
8 (2001), no. 1, Research Paper 19.
%D A000587 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.
%H A000587 T. D. Noe, <a href="b000587.txt">Table of n, a(n) for n=0..100</a>
%H A000587 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/NoteBooks/
NoteBook2/chapterIII/page5.htm">Notebook entry</a>
%H A000587 S. de Wannemacker, T. Laffey and R. Osburn, <a href="http://arXiv.org/
abs/math.NT/0608085">On a conjecture of Wilf</a>
%H A000587 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
ComplementaryBellNumber.html">Complementary Bell Number</a>
%F A000587 a(n) = e*sum(k>=0, (-1)^k*k^n/k!) - Benoit Cloitre (benoit7848c(AT)orange.fr),
Jan 28 2003
%F A000587 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.
%F A000587 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
(pauldhanna(AT)juno.com), Dec 08 2003
%F A000587 With different signs: G.f.: sum{k>=0, x^k/prod[l=1..k, 1+lx]}.
%F A000587 Recurrence: a(n) = -Sum[i=0..n-1, a(i)*C(n-1, i) ]. - Ralf Stephan, Feb
24 2005
%F A000587 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
%F A000587 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
%F A000587 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
(pbarry(AT)wit.ie), Mar 31 2008
%t A000587 Table[ -1 * Sum[ (-1)^( k + 1) StirlingS2[ n, k ], {k, 0, n} ], {n, 0,
40} ]
%o A000587 (Other) sage: expnums(26, -1)# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
May 15 2009]
%Y A000587 Cf. A000110.
%Y A000587 Cf. A011971 (base triangle PE), A078937 = PE^2
%Y A000587 A144185, A118433 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 13
2008]
%Y A000587 Sequence in context: A087042 A003678 A109322 this_sequence A014182 A131463
A065644
%Y A000587 Adjacent sequences: A000584 A000585 A000586 this_sequence A000588 A000589
A000590
%K A000587 sign,easy,nice
%O A000587 0,6
%A A000587 N. J. A. Sloane (njas(AT)research.att.com).
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