0,6

Alternating row sums of Stirling2 triangle A048993.

Related to the matrix-exponential of the Pascal-matrix, see A000110 and A011971. - Gottfried Helms (helms(AT)uni-kassel.de) 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 (qntmpkt(AT)yahoo.com), Sep 13 2008]

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

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

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.

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.

Yang, Yifan, On a multiplicative partition function, Electron. J. Combin. 8 (2001), no. 1, Research Paper 19.

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.

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

S. Ramanujan, Notebook entry

S. de Wannemacker, T. Laffey and R. Osburn, On a conjecture of Wilf

Eric Weisstein's World of Mathematics, Complementary Bell Number

a(n) = e*sum(k>=0, (-1)^k*k^n/k!) - Benoit Cloitre (benoit7848c(AT)orange.fr), 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 (pauldhanna(AT)juno.com), 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 (pbarry(AT)wit.ie), Mar 31 2008

Table[ -1 * Sum[ (-1)^( k + 1) StirlingS2[ n, k ], {k, 0, n} ], {n, 0, 40} ]

(Other) sage: expnums(26, -1)# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009]

Cf. A000110.

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

A144185, A118433 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 13 2008]

Sequence in context: A087042 A003678 A109322 this_sequence A014182 A131463 A065644

Adjacent sequences: A000584 A000585 A000586 this_sequence A000588 A000589 A000590

sign,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).