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

1, 1, 4, 5, 21, 49, 176, 513, 1720, 5401, 17777, 57421, 188657, 617177, 2033176, 6697745, 22139781, 73262233, 242931322, 806516561, 2681475049, 8925158441, 29740390673, 99196158145, 331163178476, 1106489052969, 3699881730901, 12380449027325, 41454579098853

Offset

0,3

Comments

a(n) = A238342(2n,n) = A242447(2n,n).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

Recurrence: see Maple program.

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

Example

a(1) = 1: [2].
a(2) = 4: [2,2], [1,2,1], [2,1,1], [1,1,2].
a(3) = 5: [2,2,2], [1,3,1,1], [1,1,3,1], [3,1,1,1], [1,1,1,3].
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].

Maple

a:= proc(n) option remember;
     `if`(n<5, [1, 1, 4, 5, 21][n+1],
      ((n-1)*(14911*n^4 -102036*n^3 +249203*n^2
       -252880*n +87794) *a(n-1)
      +(27528*n^5 -239548*n^4 +803564*n^3 -1283816*n^2
       +963472*n -266160) *a(n-2)
      -2*(2*n-5)*(10323*n^4 -62876*n^3 +136848*n^2
       -125584*n +40329) *a(n-3)
      +2*(2*n-7)*(n-2)*(1147*n^3 -4055*n^2 +4742*n
       -1762) *a(n-4)) / (5*(n-1)*n*
      (1147*n^3 -7496*n^2 +16293*n -11706)))
    end:
seq(a(n), n=0..35);

Mathematica

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 *)

Crossrefs

Cf. A232605, A332051.

Sequence in context: A135964 A099578 A109452 * A178625 A350827 A284911

Adjacent sequences: A232662 A232663 A232664 * A232666 A232667 A232668

Keyword

nonn

Author

Alois P. Heinz, Nov 27 2013

Status

approved