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A005121
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Ultradissimilarity relations on an n-set.
(Formerly M3649)
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5
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1, 1, 4, 32, 436, 9012, 262760, 10270696, 518277560, 32795928016, 2542945605432, 237106822506952, 26173354092593696, 3375693096567983232, 502995942483693043200, 85750135569136650473360, 16583651916595710735271248, 3611157196483089769387182064, 879518067472225603327860638128
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| First column in A154960. [From Mats Granvik (mats.granvik(AT)abo.fi), Jan 18 2009]
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REFERENCES
| L. Babai and T. Lengyel, A convergence criterion for recurrent sequences with application to the partition lattice, Analysis 12 (1992), 109-119.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 316-321.
T. Lengyel, On a recurrence involving Stirling numbers, Europ. J. Combin., 5 (1984), 313-321.
M. Schader, Hierarchical analysis: classification with ordinal object dissimilarities, Metrika, 27 (1980), 127-132.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| L. Babai and T. Lengyel, A convergence criterion for recurrent sequences with application to the partition lattice, Analysis 12 (1992), 109-119. [Bad link]
S. R. Finch, Lengyel's Constant
T. Prellberg, On the asymptotic analysis of a class of linear recurrences (slides).
Eric Weisstein's World of Mathematics, Lengyel's Constant
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FORMULA
| a(n)=Sum_{i=1..n-1} N_i(n), where N_k(m)=Sum_{j=k..m-1} Stirling2(m, j)*N_{k-1}(j), m=3..n, k=2..m-1; N_1(2)=N_1(3)=...=N_1(n)=1.
a(n) = Sum_{k=1..n-1} Stirling2(n, k)*a(k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 16 2003
E.g.f. satisfies Z(z) = 1/2 * (Z(exp(z)-1) - z). (Lengyel)
Asymptotic growth: a(n) ~ C_L*(n!)^2*(2log(2))^(-n)*n^(-1-1/3*log(2)) (Babai and Lengyel), with C_L = 1.0986858055... = A086053 (Flajolet and Salvy).
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MATHEMATICA
| a[1] = 1; a[n_] := a[n] = Sum[StirlingS2[n, k]*a[k], {k, 1, n-1}]; Array[a, 19]
(* From Jean-François Alcover, Jun 24 2011, after V. Jovovic *)
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PROG
| (PARI) {a(n) = local(A); if( n<1, 0, for(k=1, n, A = truncate(A) + x*O(x^k); A = x - A + subst(A, x, exp(x + x*O(x^k)) - 1)); n! * polcoeff(A, n))} /* Michael Somos Sep 22 2007 */
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CROSSREFS
| Cf. A006541. Row sums of A008826.
Sequence in context: A140178 A088991 A009668 * A192500 A192486 A037964
Adjacent sequences: A005118 A005119 A005120 * A005122 A005123 A005124
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 16 2003
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