A332051 Number of compositions of 2n where the multiplicity of the first part equals n.

1, 1, 3, 4, 15, 36, 126, 372, 1239, 3910, 12848, 41581, 136578, 447188, 1473342, 4855704, 16053831, 53138244, 176233968, 585202262, 1945964080, 6478043121, 21588979877, 72016891509, 240452892570, 803489258286, 2686964354376, 8991840800137, 30110638705890

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1882

Formula

a(n) = A331332(2n,n).

a(n) ~ c * d^n / sqrt(Pi*n), where d = 3.40869819984215108586487649733361214893... is the root of the equation 4 - 32*d - 8*d^2 + 5*d^3 = 0, and c = 0.34930509632919368540449993196290415079... is the root of the equation 5 - 4*c^2 - 592*c^4 + 2368*c^6 = 0. - Vaclav Kotesovec, Feb 08 2020

Recurrence: 5*(n-1)*n*(2294*n^5 - 31267*n^4 + 168064*n^3 - 445121*n^2 + 580494*n - 297864)*a(n) = (n-1)*(29822*n^6 - 415647*n^5 + 2327634*n^4 - 6668807*n^3 + 10238782*n^2 - 7910608*n + 2368800)*a(n-1) + 2*(27528*n^7 - 434848*n^6 + 2851985*n^5 - 10024036*n^4 + 20278349*n^3 - 23438626*n^2 + 14189888*n - 3420000)*a(n-2) - 2*(41292*n^7 - 647684*n^6 + 4218357*n^5 - 14743832*n^4 + 29759871*n^3 - 34533464*n^2 + 21199620*n - 5259600)*a(n-3) + 2*(n-4)*(2*n - 7)*(2294*n^5 - 19797*n^4 + 65936*n^3 - 105591*n^2 + 80846*n - 23400)*a(n-4). - Vaclav Kotesovec, Feb 08 2020

Example

a(0) = 1: the empty composition.
a(1) = 1: 2.
a(2) = 3: 22, 112, 121.
a(3) = 4: 222, 1113, 1131, 1311.
a(4) = 15: 2222, 11114, 11141, 11411, 14111, 111122, 111212, 111221, 112112, 112121, 112211, 121112, 121121, 121211, 122111.

Maple

b:= proc(n, i) option remember; `if`(n=0, x, add(expand(
     `if`(i=j, x, 1)*b(n-j, `if`(n<i+j, 0, i))), j=1..n))
    end:
a:= n-> `if`(n=0, 1, coeff(add(b(2*n-j, j), j=1..2*n), x, n)):
seq(a(n), n=0..35);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, x, Sum[Expand[If[i == j, x, 1] b[n - j, If[n < i + j, 0, i]]], {j, 1, n}]];
a[n_] := If[n == 0, 1, Coefficient[Sum[b[2 n - j, j], {j, 1, 2 n}], x, n]];
a /@ Range[0, 35] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

Crossrefs

Cf. A232665, A232697, A255197, A331332.

Sequence in context: A041665 A052133 A372342 * A209479 A209338 A127144

Adjacent sequences: A332048 A332049 A332050 * A332052 A332053 A332054

Keyword

nonn

Author

Alois P. Heinz, Feb 06 2020

Status

approved