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A005649
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Expansion of (2 - e^x)^(-2).
(Formerly M1866)
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17
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1, 2, 8, 44, 308, 2612, 25988, 296564, 3816548, 54667412, 862440068, 14857100084, 277474957988, 5584100659412, 120462266974148, 2772968936479604, 67843210855558628, 1757952715142990612, 48093560991292628228, 1385244691781856307124
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Exponential self-convolution of numbers of preferential arrangements.
Number of compatible bipartitional relations on a set of cardinality n. - Ralf Stephan, Apr 27 2003
Stirling transform of A052558 : 1, 1, 4, 12, 72, 360, . . . - Philippe DELEHAM, May 17 2005
With an extra 1 at the beginning, coefficients of the formal (divergent) series expansion at infinity of Sum_{k>=0} 1/binomial(x,k) = 1+1/x+2/x^2+8/x^3+... Also Sum_{k>=0} k!/x^k Product_{i=1..k-1} 1/(1- i/x) yields a generating function in 1/x - Roland Bacher (Roland.Bacher(AT)ujf-grenoble.fr), Nov 21 2000
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 154
Foata, D. and Krattenthaler, C., Graphical Major Indices, II, Seminaire Lotharingien de Combinatoire, B34k, 16 pp., 1995.
D. Foata and D. Zeilberger, [math/9406220] The Graphical Major Index
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FORMULA
| E.g.f.: 1/(2-exp(x))^2.
a(n) = (A000670(n) + A000670(n+1)) / 2 . - Philippe DELEHAM, May 16 2005
a(n) = D^n(1/(1-x)^2) evaluated at x = 0, where D is the operator (1+x)*d/dx. Cf. A000670 and A052841. - Peter Bala, Nov 25 2011
E.g.f.: 1/(2-exp(x))^2 = 1/(G(0) + 4), G(k) = 1-4/((2^k)-x*(4^k)/((2^k)*x-(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 15 2011
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MATHEMATICA
| f[n_] := Sum[(i + j)^n/2^(2 + i + j), {i, 0, Infinity}, {j, 0, Infinity}]; Array[f, 20, 0] [From Vladimir Reshetnikov (v.reshetnikov(AT)gmail.com), Dec 31 2008]
a[n_] := (-1)^n (PolyLog[-n-1, 2] - PolyLog[-n, 2])/4; Array[f, 20, 0] (* From Vladimir Reshetnikov, Jan 23 2011 *)
Range[0, 19]! CoefficientList[Series[(2 - Exp@ x)^-2, {x, 0, 19}], x] (* Robert G. Wilson v, Jan 23 2011 *)
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PROG
| (PARI) a(n)=if(n<0, 0, n!*polcoeff(subst(1/(1-y)^2, y, exp(x+x*O(x^n))-1), n))
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CROSSREFS
| Cf. A000670.
2*A083410(n)=a(n), if n>0.
Pairwise sums of A052841 and also of A089677.
Sequence in context: A137984 A191810 A172109 * A005363 A123307 A190818
Adjacent sequences: A005646 A005647 A005648 * A005650 A005651 A005652
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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