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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. 1
0, 0, 4, 30, 240, 2140, 21300, 235074, 2853760, 37819800, 543445380, 8416452550, 139753069104, 2476581106740, 46648575724660, 930581784937770, 19597766647728000, 434455097953799344, 10112163333554834820, 246539064280189932270, 6282671083849941925360 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

All terms are even.

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

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

EXAMPLE

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.

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-Fran├žois 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 *)

PROG

(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

(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

CROSSREFS

Extends A000262 to count conversions in addition to sets of lists.

Cf. A000041, A006152, A052852, A103194.

Sequence in context: A220727 A137971 A346579 * A213102 A052604 A038225

Adjacent sequences:  A300156 A300157 A300158 * A300160 A300161 A300162

KEYWORD

nonn

AUTHOR

Mitchell Keith Bloch, Feb 26 2018

EXTENSIONS

More terms from Mitchell Keith Bloch, Mar 05 2018

STATUS

approved

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Last modified November 28 16:42 EST 2021. Contains 349413 sequences. (Running on oeis4.)