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A001681 The partition function G(n,4).
(Formerly M1481 N0584)
11
1, 1, 2, 5, 15, 51, 196, 827, 3795, 18755, 99146, 556711, 3305017, 20655285, 135399720, 927973061, 6631556521, 49294051497, 380306658250, 3039453750685, 25120541332271, 214363100120051, 1885987611214092, 17085579637664715, 159185637725413675 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Number of '12-3 and 321-4'-avoiding permutations.
Set partitions into sets of size at most 4. The e.g.f. for partitions into sets of size at most s is exp( sum(j=1..s, x^j/j!) ). [Joerg Arndt, Dec 07 2012]
Also called restricted Stirling numbers of the second kind (see Mezo). - N. J. A. Sloane, Nov 27 2013
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..609 (terms 0..200 from Alois P. Heinz)
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Filippo Disanto and Thomas Wiehe, Some instances of a sub-permutation problem on pattern avoiding permutations, arXiv preprint arXiv:1210.6908 [math.CO], 2012.
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
T. Mansour, Restricted permutations by patterns of type 2-1, arXiv:math/0202219 [math.CO], 2002.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013 and J. Int. Seq. 17 (2014) #14.1.1 .
F. L. Miksa, L. Moser and M. Wyman, Restricted partitions of finite sets, Canad. Math. Bull., 1 (1958), 87-96.
FORMULA
E.g.f.: exp( x + x^2/2 + x^3/6 + x^4/24 ). - Ralf Stephan, Apr 22 2004
a(n) = n! * sum(k=1..n, 1/k! * sum(j=0..k, C(k,j) * sum(i=j..n-k+j, C(j,i-j) * C(k-j,n-3*k+3*j-i) * 2^(5*k-4*j+i-2*n) * 3^(j-k)))). [Vladimir Kruchinin, Jan 25 2011]
a(n) = G(n,4) with G(0,i) = 1, G(n,i) = 0 for n>0 and i<1, otherwise G(n,i) = Sum_{j=0..floor(n/i)} G(n-i*j,i-1) * n!/(i!^j*(n-i*j)!*j!). - Alois P. Heinz, Apr 20 2012
Recurrence: 6*a(n) = 6*a(n-1) + 6*(n-1)*a(n-2) + 3*(n-2)*(n-1)*a(n-3) + (n-3)*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Sep 15 2013
a(n) ~ n^(3*n/4)*exp(31*(6*n)^(1/4)/64 + 5*sqrt(6*n)/16 + (6*n)^(3/4)/6 - 3*n/4 - 21/32)/(2*6^(n/4)) * (1 + 1599*6^(3/4)/(40960*n^(1/4)) + 280873603/1677721600*sqrt(6/n) + 33870741297579 /240518168576000 *6^(1/4)/n^(3/4)). - Vaclav Kotesovec, Sep 15 2013
MAPLE
G:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(G(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
end:
a:= n-> G(n, 4):
seq(a(n), n=0..30); # Alois P. Heinz, Apr 20 2012
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
a(n-i)*binomial(n-1, i-1), i=1..min(n, 4)))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Sep 22 2016
# Recurrence:
rec := {(-n^3-6*n^2-11*n-6)*f(n) + (-3*n^2-15*n-18)*f(n+1) + (-6*n-18)*f(n+2) - 6*f(n+3) + 6*f(n+4)=0, f(0)=1, f(1)=1, f(2)=2, f(3)=5}:
aList := gfun:-rectoproc(rec, f(n), list): aList(24); # Peter Luschny, Feb 26 2018
MATHEMATICA
g[n_, k_] := g[n, k] = If[n == 0, 1, If[k<1, 0, Sum[g[n-k*j, k-1]*n!/k!^j/(n-k*j)!/j!, {j, 0, n/k}]]]; Table[g[n, 4], {n, 0, 24}] (* Jean-François Alcover, Mar 11 2014, after Alois P. Heinz *)
PROG
(PARI) A001681(n)=n!*sum(k=1, n, 1/k!*sum(j=0, k, binomial(k, j)*sum(i=j, n-k+j, binomial(j, i-j)*binomial(k-j, n-3*k+3*j-i)*2^(5*k-4*j+i-2*n)*3^(j-k))));
vector(33, n, A001681(n-1)) /* Joerg Arndt, Jan 25 2011 */
(PARI) x='x+O('x^66); Vec(serlaplace(exp(sum(j=1, 4, x^j/j!)))) \\ Joerg Arndt, Mar 11 2014
CROSSREFS
Column k=4 of A229223.
Sequence in context: A287253 A117426 A201168 * A343665 A192553 A053553
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Ralf Stephan, Apr 22 2004
STATUS
approved

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Last modified March 19 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)