A263897 Number of anagram compositions of 2n that can be formed from the compositions of n.

1, 1, 2, 6, 16, 44, 134, 414, 1290, 4102, 13306, 43718, 145176, 487384, 1651154, 5636702, 19381392, 67074420, 233483430, 817106622, 2873589060, 10151255976, 36009769278, 128231072994, 458268615966, 1643227382022, 5910606009330, 21322486928518, 77132729929864

Offset: 0

Author

Gregory L. Simay, Oct 28 2015

Keywords

• nonn

Comments

A composition of n is an ordered sequence of positive integers whose sum is n. An anagram composition of n can be divided into two consecutive subsequences having the same parts, with a central part between the subsequences permitted.

a(n) is the number of anagram compositions of 2n with no central summand.

Here is a computation algorithm: List the individual partitions of n. Then for each partition, determine the corresponding number of compositions. Square each of these numbers then add them together.

Examples

			For n=3, the compositions are [3], [1,2], [2,1], [1,1,1]. The anagram compositions of 2*3=6 are [3][3], [1,2][1,2], [1,2][2,1], [2,1][1,2], [2,1][2,1], [1,1,1][1,1,1], so there are a(3)=6 anagram compositions in all.
For n=4, the individual partitions are [4], [3,1], [2,2], [2,1,1] and [1,1,1,1]. The corresponding number of compositions are 1, 2, 1, 3, and 1. The corresponding squares of these numbers are 1, 4, 1, 9, and 1. The sum of these squares is a(4)=16.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Miklos Bona, Rebecca Smith, An Involution on Involutions and a Generalization of Layered Permutations, arXiv preprint arXiv:1605.06158 [math.CO], 2016. See Theorem 4.3.

Crossrefs

First differences of A097085.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(n=0, p!^2,
      `if`(i<1, 0, add(b(n-i*j, i-1, p+j)/j!^2, j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Oct 29 2015

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!^2, If[i<1, 0, Sum[b[n-i*j, i-1, p+j]/j!^2, {j, 0, n/i}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Nov 05 2015, after Alois P. Heinz *)

Python

from sympy.core.cache import cacheit
from sympy import factorial as f
@cacheit
def b(n, i, p): return f(p)**2 if n==0 else 0 if i<1 else sum(b(n - i*j, i - 1, p + j)//f(j)**2 for j in range(n//i + 1))
def a(n): return b(n, n, 0)
print([a(n) for n in range(36)]) # Indranil Ghosh, Aug 18 2017, after Maple code

Extensions

a(9)-a(28) from Alois P. Heinz, Oct 29 2015