A001681 The partition function G(n,4).
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
Offset: 0
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.
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 19
- 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.
Programs
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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 *) -
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 */
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PARI
x='x+O('x^66); Vec(serlaplace(exp(sum(j=1,4,x^j/j!)))) \\ Joerg Arndt, Mar 11 2014
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
Extensions
More terms from Ralf Stephan, Apr 22 2004
Comments