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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
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
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: A372086 A362245 A362237 * A052867 A226238 A179495
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