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A130495 Number of compositions of 2n in which each part has even multiplicity.

%I A130495 #48 Oct 27 2023 21:52:43
%S A130495 1,1,2,8,24,72,264,952,3352,11960,43656,160840,594568,2215480,8300056,
%T A130495 31191480,117674504,445439944,1691011464,6437425720,24564925848,
%U A130495 93937631544,359943235080,1381706541512,5312678458888,20458827990456,78898261863832,304666752525368
%N A130495 Number of compositions of 2n in which each part has even multiplicity.
%C A130495 Consider the compositions of n that are capable of being rearranged into a palindrome, with a fixed, central summand allowed. Then the number of such palindrome-capable compositions of 2n or 2n+1 is a(0)+...+a(n). - _Gregory L. Simay_, Nov 27 2018
%H A130495 Alois P. Heinz, <a href="/A130495/b130495.txt">Table of n, a(n) for n = 0..1000</a> (first 51 terms from Vladeta Jovovic)
%F A130495 a(n) ~ 2^(2*n-1) / n. - _Vaclav Kotesovec_, Sep 10 2014
%e A130495 a(3) = 8 because we have: 3+3, 2+2+1+1, 2+1+2+1, 2+1+1+2, 1+2+2+1, 1+2+1+2, 1+1+2+2, 1+1+1+1+1+1. - _Geoffrey Critzer_, May 12 2014
%e A130495 Note that in Geoffrey's example (in which there's no central summand) all 8 compositions of 6=3*2 are either palindromes or can be rearranged into palindromes. The compositions of 2*2=4 with even multiplic1ty are 2+2 and 1+1+1+1, and are counted by a(2). Adding a fixed, central summand of 2, yields 2 more palindrome-capable compositions of 6: 2+2+2 and 1+1+2+1+1. The composition of 2*1=2 with even multiplicity is 1+1. Adding a fixed, central summand of 4 yields 1 more palindrome composition of 6: 1+4+1. Finally, the bare central summand of 6 is counted by a(0)=1. Hence, the total number of  compositions of 6 that are palindrome capable is a(0)+...+a(3), if the central summand is fixed. This sum also gives the total number of palindrome-capable compositions of 7, employing fixed, central summands of 1,3,5 and 7. _Gregory L. Simay_, Nov 27 2018
%p A130495 b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
%p A130495       `if`(irem(j, 2)=0, b(n-i*j, i-1, p+j)/j!, 0), j=0..n/i)))
%p A130495     end:
%p A130495 a:= n-> b(2*n$2, 0):
%p A130495 seq(a(n), n=0..35);  # _Alois P. Heinz_, May 12 2014
%t A130495 Select[Table[Length[Select[Level[Map[Permutations,IntegerPartitions[n]],{2}],Apply[And,EvenQ[Table[Count[#,#[[i]]],{i,1,Length[#]}]]]&]],{n,0,20}],#>0&] (* _Geoffrey Critzer_, May 12 2014 *)
%t A130495 b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0, Sum[If[Mod[j, 2] == 0, b[n - i*j, i - 1, p + j]/j!, 0], {j, 0, n/i}]]];
%t A130495 a[n_] := b[2n, 2n, 0];
%t A130495 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Aug 30 2016, after _Alois P. Heinz_ *)
%Y A130495 Cf. A242391 (for odd multiplicity).
%K A130495 nonn
%O A130495 0,3
%A A130495 _Vladeta Jovovic_, Aug 08 2007