A052887 - OEIS

%I #64 Sep 16 2021 06:18:37

%S 1,0,2,12,84,720,7320,85680,1130640,16571520,266747040,4673592000,

%T 88476252480,1798674958080,39061703640960,902110060051200,

%U 22068313153286400,569874634276147200,15486794507222438400,441703937156940057600,13189422568491333964800,411420697666247453184000

%N Expansion of e.g.f.: exp(x^2/(1 - x)^2).

%C Previous name was: A simple grammar.

%C 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

%H Seiichi Manyama, <a href="/A052887/b052887.txt">Table of n, a(n) for n = 0..433</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=860">Encyclopedia of Combinatorial Structures 860</a>

%F E.g.f.: exp(x^2/(1 - x)^2).

%F 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.

%F a(n) = Sum_{k=0..floor(n/2)} n!/k!*binomial(n-1, 2*k-1). - _Vladeta Jovovic_, Sep 13 2003

%F 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

%F 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

%e 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

%p spec := [S,{B=Sequence(Z,1 <= card),C=Prod(B,B),S= Set(C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%t nn = 20; a = x/(1 - x); Range[0, nn]! CoefficientList[Series[Exp[ a^2], {x, 0, nn}], x] (* _Geoffrey Critzer_, Dec 11 2011 *)

%o (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

%o (PARI)

%o N=33;  x='x+O('x^N);

%o egf=exp(x^2/(1-x)^2);

%o Vec(serlaplace(egf))

%o /* _Joerg Arndt_, Sep 15 2012 */

%K easy,nonn

%O 0,3

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000

%E New name using e.g.f. from _Vaclav Kotesovec_, Oct 01 2013

%E Formula section edited by _Petros Hadjicostas_, Jun 12 2020