A003011
Number of permutations of up to n kinds of objects, where each kind of object can occur at most two times.
Original entry on oeis.org
1, 3, 19, 271, 7365, 326011, 21295783, 1924223799, 229714292041, 35007742568755, 6630796801779771, 1527863209528564063, 420814980652048751629, 136526522051229388285611
Offset: 0
- J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 17.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
a(n) = Sum[C(n, k)*
A105749(k), 0<=k<=n]
Replace "sequence" with "collection" in comment:
A105748.
Replace "sets" with "lists" in comment:
A082765.
-
Table[nn=2n;a=1+x+x^2/2!;Total[Range[0,nn]!CoefficientList[Series[a^n,{x,0,nn}],x]],{n,0,15}] (* Geoffrey Critzer, Dec 23 2011 *)
-
a(n)=local(A);if(n<0,0,A=(1+x+x^2/2)^n;sum(k=0,2*n,k!*polcoeff(A,k)))
A099022
a(n) = Sum_{k=0..n} C(n,k)*(2*n-k)!.
Original entry on oeis.org
1, 3, 38, 1158, 65304, 5900520, 780827760, 142358474160, 34209760152960, 10478436416945280, 3984884716852972800, 1842169367191937414400, 1017403495472574045158400, 661599650478455071589606400, 500354503197888042597961267200, 435447353708763072625260119808000
Offset: 0
- Robert Israel, Table of n, a(n) for n = 0..224
- L. I. Nicolaescu, Derangements and asymptotics of the Laplace transforms of large powers of a polynomial, New York J. Math. 10 (2004) 117-131.
- Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
- P. J. Rossky, M. Karplus, The enumeration of Goldstone diagrams in many-body perturbation theory, J. Chem. Phys. 64 (1976) 1569, equation (9).
- Index entries for related partition-counting sequences
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f:= gfun:-rectoproc({a(n)=2*n*(2*n-1)*a(n-1)+n*(n-1)*a(n-2), a(0)=1,a(1)=3},a(n),remember):
map(f, [$0..20]); # Robert Israel, Feb 15 2017
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Table[(2k)! Hypergeometric1F1[-k, -2k, 1], {k, 0, 10}] (* Vladimir Reshetnikov, Feb 16 2011 *)
Table[Sum[Binomial[n,k](2n-k)!,{k,0,n}],{n,0,20}] (* Harvey P. Dale, Nov 22 2021 *)
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for(n=0,25, print1(sum(k=0,n, binomial(n,k)*(2*n-k)!), ", ")) \\ G. C. Greubel, Dec 31 2017
A105747
Number of ways to use the elements of {1,..,k}, 0<=k<=2n, once each to form a collection of n (possibly empty) lists, each of length at most 2.
Original entry on oeis.org
1, 4, 23, 216, 2937, 52108, 1136591, 29382320, 877838673, 29753600404, 1127881002535, 47278107653768, 2171286661012617, 108417864555606300, 5847857079417024031, 338841578119273846112
Offset: 0
Robert A. Proctor (www.math.unc.edu/Faculty/rap/), Apr 18 2005
a(2)=23:
{(),()},
{(),(1)},
{(),(1,2)},
{(),(2,1)},
{(1),(2)},
{(1),(2,3)},
{(1),(3,2)},
...,
{(1,4),(2,3)},
{(1,4),(3,2)},
{(4,1),(2,3)},
{(4,1),(3,2)}.
Replace "collection" by "sequence":
A082765.
Replace "lists" by "sets":
A105748.
-
Table[Sum[(k+i)!/i!/(k-i)!, {k, 0, n}, {i, 0, k}], {n, 0, 20}]
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