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A232665 Number of compositions of 2n such that the largest multiplicity of parts equals n. 5
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 (list; graph; refs; listen; history; text; internal format)
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 A284911 A091130

Adjacent sequences:  A232662 A232663 A232664 * A232666 A232667 A232668

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Nov 27 2013

STATUS

approved

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Last modified December 4 18:08 EST 2021. Contains 349526 sequences. (Running on oeis4.)