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 A000248 Expansion of e.g.f. exp(x*exp(x)). (Formerly M2857 N1148) 95
 1, 1, 3, 10, 41, 196, 1057, 6322, 41393, 293608, 2237921, 18210094, 157329097, 1436630092, 13810863809, 139305550066, 1469959371233, 16184586405328, 185504221191745, 2208841954063318, 27272621155678841, 348586218389733556, 4605223387997411873 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Number of forests with n nodes and height at most 1. Equivalently, number of idempotent mappings f from a set of n elements into itself (i.e., satisfying f o f = f). - Robert FERREOL, Oct 11 2007 In other words, a(n) = number of idempotents in the full semigroup of maps from [1..n] to itself. [Tainiter] a(n) is the number of ways to select a set partition of {1,2,...,n} and then designate one element in each block (cell) of the partition. Let set B have cardinality n. Then a(n) is the number of functions f:D->C over all partitions {D,C} of B. See the example in the Example Section below. We note that f:empty set->B is designated as the null function, whereas f:B->empty set is undefined unless B itself is empty. - Dennis P. Walsh, Dec 05 2013 In physics, a(n) would be interpreted as the number of projection operators P on S_n, i.e., ones satisfying P^2 = P. Example: a particle with a half-integer spin s has a spin space with 2s+1 base states which admits a(2s+1) linear projection operators (including the identity). These are important because they satisfy the operator identity exp(zU) = 1+(exp(z)-1)*U, valid for any complex z. - Stanislav Sykora, Nov 03 2016 REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91. 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). R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.32(d). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..541 (first 101 terms from T. D. Noe) Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021. P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 131. Xing Gao and William F. Keigher, Interlacing of Hurwitz series, Communications in Algebra, 45:5 (2017), 2163-2185, DOI: 10.1080/00927872.2016.1226885. See Ex. 2.13. B. Harris and L. Schoenfeld, The number of idempotent elements in symmetric semigroups, J. Combin. Theory, 3 (1967), 122-135. Bernard Harris and Lowell Schoenfeld, Asymptotic expansions for the coefficients of analytic functions, Illinois Journal of Mathematics, Volume 12, Issue 2 (1968), 264-277. G. Helms, Pascalmatrix tetrated [From Gottfried Helms, Feb 04 2009] INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 117 Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5. J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103. J. Riordan, Letter to N. J. A. Sloane, Oct. 1970 John Riordan and N. J. A. Sloane, Correspondence, 1974 Emre Sen, Exceptional Sequences and Idempotent Functions, arXiv:1909.05887 [math.RT], 2019. M. Tainiter, A characterization of idempotents in semigroups, J. Combinat. Theory, 5 (1968), 370-373. Haoliang Wang and Robert Simon, The Analysis of Synchronous All-to-All Communication Protocols for Wireless Systems, Q2SWinet'18: Proceedings of the 14th ACM International Symposium on QoS and Security for Wireless and Mobile Networks (2018), 39-48. FORMULA G.f.: Sum_{k>=0} x^k/(1-k*x)^(k+1). - Vladeta Jovovic, Oct 25 2003 a(n) = Sum_{k=0..n} C(n,k)*(n-k)^k. - Paul D. Hanna, Jun 26 2009 G.f.: G(0) where G(k) = 1 - x*(-1+2*k*x)^(2*k+1)/((x-1+2*k*x)^(2*k+2) - x*(x-1+2*k*x)^(4*k+4)/(x*(x-1+2*k*x)^(2*k+2) - (2*x-1+2*k*x)^(2*k+3)/G(k+1))) (continued fraction). - Sergei N. Gladkovskii, Jan 26 2013 E.g.f.: 1 + x/(1+x)*(G(0) - 1) where G(k) = 1 + exp(x)/(k+1)/(1-x/(x+(1)/G(k+1))) (continued fraction). - Sergei N. Gladkovskii, Feb 04 2013 Recurrence: a(0)=1, a(n) = Sum_{k=1..n} binomial(n-1,k-1)*k*a(n-k). - James East, Mar 30 2014 Asymptotics (Harris and Schoenfeld, 1968): a(n) ~ sqrt((r+1)/(2*Pi*(n+1)*(r^2+3*r+1))) * n! * exp((n+1)/(r+1)) / r^n, where r is the root of the equation r*(r+1)*exp(r) = n+1. - Vaclav Kotesovec, Jul 13 2014 a(n) = Sum_{k=0..n} A005727(k)*Stirling2(n, k). - Mélika Tebni, Jun 12 2022 EXAMPLE a(3)=10 since, for B={1,2,3}, we have 10 functions: 1 function of the type f:empty set->B; 6 functions of the type f:{x}->B\{x}; and 3 functions of the type f:{x,y}->B\{x,y}. - Dennis P. Walsh, Dec 05 2013 MAPLE A000248 := proc(n) local k; add(k^(n-k)*binomial(n, k), k=0..n); end; # Robert FERREOL, Oct 11 2007 a:= proc(n) option remember; if n=0 then 1 else add(binomial(n-1, j) *(j+1) *a(n-1-j), j=0..n-1) fi end: seq(a(n), n=0..20); # Zerinvary Lajos, Mar 28 2009 MATHEMATICA CoefficientList[Series[Exp[x Exp[x]], {x, 0, 20}], x]*Table[n!, {n, 0, 20}] a = 1; a = 1; a[n_] := a[n] = a[n - 1] + Sum[(Binomial[n - 1, j] + (n - 1) Binomial[n - 2, j]) a[j], {j, 0, n - 2}]; Table[a[n], {n, 0, 20}] (* David Callan, Oct 04 2013 *) Flatten[{1, Table[Sum[Binomial[n, k]*(n-k)^k, {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jul 13 2014 *) Table[Sum[BellY[n, k, Range[n]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *) PROG (PARI) a(n)=sum(k=0, n, binomial(n, k)*(n-k)^k); \\ Paul D. Hanna, Jun 26 2009 (PARI) x='x+O('x^66); Vec(serlaplace(exp(x*exp(x)))) \\ Joerg Arndt, Oct 06 2013 (Sage) # uses[bell_matrix from A264428] B = bell_matrix(lambda k: k+1, 20) print([sum(B.row(n)) for n in range(20)]) # Peter Luschny, Sep 03 2019 (Magma) m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Feb 01 2020 CROSSREFS First row of array A098697. Row sums of A133399. Column k=1 of A210725, A279636. Column k=2 of A245501. Cf. A005727, A048993. Sequence in context: A325059 A116540 A236407 * A245504 A305405 A030927 Adjacent sequences:  A000245 A000246 A000247 * A000249 A000250 A000251 KEYWORD easy,nonn,nice AUTHOR EXTENSIONS In view of the multiple appearances of this sequence, I replaced the definition with the simple exponential generating function. - N. J. A. Sloane, Apr 16 2018 STATUS approved

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Last modified October 2 17:06 EDT 2022. Contains 357227 sequences. (Running on oeis4.)