A255197 -id:A255197 - cp's OEIS frontend

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

Alois P. Heinz, Feb 06 2020

nonn

			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.

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

Cf. A232665, A232697, A255197, A331332.

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);

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 *)

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

A253192 Number of ways to place nonintersecting diagonals in convex (n+3)-gon so as to create exactly one triangle.

Original entry on oeis.org

1, 0, 5, 6, 35, 80, 309, 890, 3058, 9580, 31863, 103054, 340415, 1116032, 3688745, 12176814, 40344505, 133742500, 444262378, 1477142040, 4918099660, 16390294664, 54679621775, 182572812266, 610115196150, 2040383498748, 6828408179435, 22866979920390, 76623655367703, 256899191586880, 861774049296325

Offset: 0

Michael D. Weiner, Mar 24 2015

nonn

			a(1)=0 since there are no dissections of a convex quadrilateral with exactly one triangle.
a(2)=5 because we can place one diagonal in a pentagon 5 different ways, each time creating one triangle and one quadrilateral.

D. Birmajer, J. B. Gil, and M. Weiner, Colored partitions of a convex polygon by noncrossing diagonals, arXiv:1503.05242 [math.CO], 2015.

Cf. A255197.

a:=n->sum(binomial(n+k+2, k)*binomial(n-k-1, k-1), k = 1 .. trunc((1/2)*n)): (1, seq(a(n), n=1..30));
ogf := (RootOf((4*x^3-32*x^2-8*x+5)*_Z^3+(-9*x^4+42*x^3+249*x^2+96*x-51)*_Z+18*x^4-116*x^3-269*x^2-128*x+62)-2)/(3*x^2);
gfun[seriestolist](series(ogf, x=0, 30))[]; # Mark van Hoeij, Nov 28 2024

Prepend[Table[Sum[Binomial[n + k + 2, k]*Binomial[n - k - 1, k - 1], {k, 1, n/2}], {n, 1, 30}], 1] (* Michael De Vlieger, Mar 24 2015 *)

a(n) = Sum_{k=1..floor(n/2)} C(n+k+2,k)*C(n-k-1,k-1), n>0.
D-finite with recurrence: 0=2*(n-1)*(2*n-3)*(n+1)*(37*n^3 + 97*n^2 + 76*n + 20)*a(n-3) - 2*n*(592*n^5 + 960*n^4 - 15*n^3 - 70*n^2 + 263*n + 70)*a(n-2) - 2*n*(n-1)*(n+1)*(148*n^3 + 314*n^2 + 37*n - 89)*a(n-1) + 5*n*(n+2)*(n+1)*(37*n^3 - 14*n^2 - 7*n + 4)*a(n).
a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = 3.4086981998421510858648764973336... is the real root of the equation 4 - 32*d - 8*d^2 + 5*d^3 = 0 and c = 0.8203071528123829561131676776610304796... is the smallest positive real root of the equation 1 + 402019*c - 584933*c^2 + 115625*c^3 = 0. - Vaclav Kotesovec, Jul 05 2024

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

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