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 A052887 Expansion of e.g.f.: exp(x^2/(1 - x)^2). 11
 1, 0, 2, 12, 84, 720, 7320, 85680, 1130640, 16571520, 266747040, 4673592000, 88476252480, 1798674958080, 39061703640960, 902110060051200, 22068313153286400, 569874634276147200, 15486794507222438400, 441703937156940057600, 13189422568491333964800, 411420697666247453184000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Previous name was: A simple grammar. For n>=2, a(n) is the number of ways to partition {1,2,...,n} into any number of blocks. Then partition each block into exactly 2 sub-blocks. Then form ordered pairs by permuting the elements within each pair of sub-blocks. - Geoffrey Critzer, Jun 13 2020 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..433 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 860 FORMULA E.g.f.: exp(x^2/(1 - x)^2). Recurrence: a(0) = 1, a(1) = 0, a(2) = 2, and for n >= 2, (-n^3-2*n-3*n^2)*a(n) +(3*n^2+7*n+2)*a(n+1) + (-6-3*n)*a(n+2) + a(n+3) = 0. a(n) = Sum_{k=0..floor(n/2)} n!/k!*binomial(n-1, 2*k-1). - Vladeta Jovovic, Sep 13 2003 a(n) ~ 2^(1/6)* n^(n-1/6) * exp(1/3 - (n/2)^(1/3) + 3*(n/2)^(2/3) - n)/sqrt(3) * (1 - 14*2^(-2/3)/(27*n^(1/3)) - 1688*2^(-4/3)/(3645*n^(2/3))). - Vaclav Kotesovec, Oct 01 2013 a(n) = n!*y(n) with y(0) = 1 and y(n) = Sum_{k=0..n-1} (n-k)*(n-k-1)*y(k)/n for n >= 1. - Benedict W. J. Irwin, Jun 02 2016 EXAMPLE a(3) = 12 because we have the 6 ordered pairs: ({1},{2,3}), ({1},{3,2}), ({2},{1,3}), ({2},{3,1}), ({3},{1,2}), ({3},{2,1}) and their reflections. - Geoffrey Critzer, Jun 13 2020 MAPLE spec := [S, {B=Sequence(Z, 1 <= card), C=Prod(B, B), S= Set(C)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20); MATHEMATICA nn = 20; a = x/(1 - x); Range[0, nn]! CoefficientList[Series[Exp[ a^2], {x, 0, nn}], x] (* Geoffrey Critzer, Dec 11 2011 *) PROG (Maxima) makelist(if n=0 then 1 else sum(n!/k!*binomial(n-1, 2*k-1), k, 0, floor(n/2)), n, 0, 18); \\ Bruno Berselli, May 25 2011 (PARI) N=33; x='x+O('x^N); egf=exp(x^2/(1-x)^2); Vec(serlaplace(egf)) /* Joerg Arndt, Sep 15 2012 */ CROSSREFS Sequence in context: A235351 A362245 A362237 * A052867 A226238 A179495 Adjacent sequences: A052884 A052885 A052886 * A052888 A052889 A052890 KEYWORD easy,nonn AUTHOR encyclopedia(AT)pommard.inria.fr, Jan 25 2000 EXTENSIONS New name using e.g.f. from Vaclav Kotesovec, Oct 01 2013 Formula section edited by Petros Hadjicostas, Jun 12 2020 STATUS approved

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Last modified June 5 16:06 EDT 2023. Contains 363137 sequences. (Running on oeis4.)