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A232665 Number of compositions of 2n such that the largest multiplicity of parts equals n.

%I A232665 #22 Feb 06 2020 23:02:05
%S A232665 1,1,4,5,21,49,176,513,1720,5401,17777,57421,188657,617177,2033176,
%T A232665 6697745,22139781,73262233,242931322,806516561,2681475049,8925158441,
%U A232665 29740390673,99196158145,331163178476,1106489052969,3699881730901,12380449027325,41454579098853
%N A232665 Number of compositions of 2n such that the largest multiplicity of parts equals n.
%C A232665 a(n) = A238342(2n,n) = A242447(2n,n).
%H A232665 Alois P. Heinz, <a href="/A232665/b232665.txt">Table of n, a(n) for n = 0..1000</a>
%F A232665 Recurrence: see Maple program.
%F A232665 a(n) ~ c*r^n/sqrt(Pi*n), where r = 3.408698199842151... is the root of the equation 4 - 32*r - 8*r^2 + 5*r^3 = 0 and c = 0.479880052557486135... is the root of the equation 1 + 384*c^2 - 2368*c^4 + 2960*c^6 = 0. - _Vaclav Kotesovec_, Nov 27 2013
%e A232665 a(1) = 1: [2].
%e A232665 a(2) = 4: [2,2], [1,2,1], [2,1,1], [1,1,2].
%e A232665 a(3) = 5: [2,2,2], [1,3,1,1], [1,1,3,1], [3,1,1,1], [1,1,1,3].
%e A232665 a(4) = 21: [2,2,2,2], [1,1,4,1,1], [4,1,1,1,1], [1,4,1,1,1], [1,1,1,4,1], [1,1,1,1,4], [1,2,1,1,1,2], [2,1,1,1,1,2], [2,1,2,1,1,1], [1,2,2,1,1,1],[1,2,1,2,1,1], [2,1,1,2,1,1], [1,2,1,1,2,1], [2,1,1,1,2,1],[1,1,2,1,2,1], [1,1,2,2,1,1], [2,2,1,1,1,1], [1,1,1,2,2,1], [1,1,2,1,1,2], [1,1,1,2,1,2], [1,1,1,1,2,2].
%p A232665 a:= proc(n) option remember;
%p A232665      `if`(n<5, [1, 1, 4, 5, 21][n+1],
%p A232665       ((n-1)*(14911*n^4 -102036*n^3 +249203*n^2
%p A232665        -252880*n +87794) *a(n-1)
%p A232665       +(27528*n^5 -239548*n^4 +803564*n^3 -1283816*n^2
%p A232665        +963472*n -266160) *a(n-2)
%p A232665       -2*(2*n-5)*(10323*n^4 -62876*n^3 +136848*n^2
%p A232665        -125584*n +40329) *a(n-3)
%p A232665       +2*(2*n-7)*(n-2)*(1147*n^3 -4055*n^2 +4742*n
%p A232665        -1762) *a(n-4)) / (5*(n-1)*n*
%p A232665       (1147*n^3 -7496*n^2 +16293*n -11706)))
%p A232665     end:
%p A232665 seq(a(n), n=0..35);
%t A232665 b[n_, s_] := b[n, s] = If[n == 0, 1, If[n<s, 0, Expand[Sum[b[n-j, s]*x, {j, s, n}]]]]; T[n_, k_] := If[k == 0, If[n == 0, 1, 0], Sum[Function[{p}, Sum[ Coefficient[p, x, i]*Binomial[i+k, k], {i, 0, Exponent[p, x]}]][b[n-j*k, j+1]], {j, 1, n/k}]]; a[n_] := T[2n, n]; Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Feb 09 2015, after A238342 *)
%Y A232665 Cf. A232605, A332051.
%K A232665 nonn
%O A232665 0,3
%A A232665 _Alois P. Heinz_, Nov 27 2013