A300159 Number of ways of converting one set of lists containing n elements to another set of lists containing n elements by removing the last element from one of the lists and either appending it to an existing list or treating it as a new list.
0, 0, 4, 30, 240, 2140, 21300, 235074, 2853760, 37819800, 543445380, 8416452550, 139753069104, 2476581106740, 46648575724660, 930581784937770, 19597766647728000, 434455097953799344, 10112163333554834820, 246539064280189932270, 6282671083849941925360
Offset: 0
Keywords
Examples
a(0) = 0 since for 0 lists, 0 conversions are possible. a(1) = 0 since for the 1 set of 1 list of length 1, there exist no possible conversions. a(2) = 4 since for the 2 sets of 1 list of length 2, there exists only 1 conversion, and for the 1 set of 2 lists of length 1, there exist 2 conversions. a(3) = 30 since for the 6 sets of 1 list of length 3, there exists 1 conversion, for the 6 sets of 1 list of length 2 and 1 list of length 1, there exist 3 conversions, and for the 1 set of 3 lists of length 1, there exists 6 conversions.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..443 (first 71 terms from Mitchell Keith Bloch)
- Mitchell Keith Bloch, Program C++
- Mitchell Keith Bloch, Program C++ with Boost
- Index entries for sequences related to Laguerre polynomials
Crossrefs
Programs
-
Magma
l:= func< n,b | Evaluate(LaguerrePolynomial(n,1), b) >; [0,0,4]cat[Factorial(n)*( 2*l(n-2,-1) - l(n-3,-1) ): n in [3..30]]; // G. C. Greubel, Mar 09 2021
-
Maple
b:= proc(n, t, c) option remember; `if`(n=0, t^2-c, add(j!* binomial(n-1, j-1)*b(n-j, t+1, c+`if`(j=1, 1, 0)), j=1..n)) end: a:= n-> b(n, 0$2): seq(a(n), n=0..25); # Alois P. Heinz, Mar 05 2018 # second Maple program: a:= proc(n) option remember; `if`(n<6, [0$2, 4, 30, 240, 2140][n+1], (n*(2*n^2-13*n+16)*a(n-1)-n*(n-1)*(n-3)*(n-4)*a(n-2))/((n-2)*(n-5))) end: seq(a(n), n=0..25); # Alois P. Heinz, Mar 05 2018 -
Mathematica
(* First program *) b[n_, t_, c_]:= b[n,t,c]= If[n==0, t^2 -c, Sum[j! Binomial[n-1, j-1]b[n-j,t+1,c + If[j==1, 1, 0]], {j,n}]]; a[n_]:= b[n, 0, 0]; a/@ Range[0, 25] (* Jean-François Alcover, Nov 24 2020, after Alois P. Heinz *) (* Second program *) Table[If[n<2, 0, n!*(2*LaguerreL[n-2,1,-1] -LaguerreL[n-3,1,-1])], {n,0,30}] (* G. C. Greubel, Mar 09 2021 *) -
Sage
[0,0,4]+[factorial(n)*(2*gen_laguerre(n-2,1,-1) - gen_laguerre(n-3,0,-1)) for n in (3..30)] # G. C. Greubel, Mar 09 2021
Formula
a(n) = Sum_{i=1..p(n)} (n!/(Product_{j=1..n} k(i,j)!) * ((Sum_{j=1..n} k(i,j))^2 - k(i,1))) (where p(n) is the number of partitions A000041 and k(i,j) is the number of partitions of size j in partitioning i).
From Alois P. Heinz, Mar 05 2018: (Start)
E.g.f.: x^2*(2-x)*exp(x/(1-x))/(x-1)^2.
a(n) = (n*(2*n^2-13*n+16)*a(n-1) - n*(n-1)*(n-3)*(n-4)*a(n-2)) / ((n-2)*(n-5)) for n>5. (End)
a(n) ~ n^(n + 3/4) * exp(2*sqrt(n) - n - 1/2) / sqrt(2). - Vaclav Kotesovec, Jun 02 2018
a(n) = n!*( 2*LaguerreL(n-2,1,-1) - LaguerreL(n-3,1,-1) ) for n > 1, with a(0) = a(1) = 0. - G. C. Greubel, Mar 09 2021
Extensions
More terms from Mitchell Keith Bloch, Mar 05 2018
Comments