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

Author

Alois P. Heinz, Feb 06 2020

Keywords

• nonn

Examples

			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.
		

Links

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

Crossrefs

Cf. A232665, A232697, A255197, A331332.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, x, add(expand(
     `if`(i=j, x, 1)*b(n-j, `if`(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 *)

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