A272493 Number of ordered set partitions of [n] with nondecreasing block sizes and maximal block size equal to three.

1, 4, 30, 200, 1610, 13440, 130200, 1327200, 15107400, 183321600, 2422820400, 34104470400, 515897382000, 8276556288000, 141290381232000, 2546760408192000, 48489153817104000, 970454450085120000, 20400874234060320000, 448974320483969280000

Offset: 3

Author

Alois P. Heinz, May 01 2016

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..450

Crossrefs

Column k=3 of A262071.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1)+`if`(i>n, 0, binomial(n, i)*b(n-i, i))))
    end:
a:= n-> (k-> b(n, k) -b(n, k-1))(3):
seq(a(n), n=3..30);

Mathematica

FullSimplify[Table[n! * ((-35*(1 + Sqrt[2]) + 7*2^(1 + n/2)* (3*Sqrt[2] - 2) - 5*(-1)^n*(17*Sqrt[2] - 23))/2^(n/2) + 2^(5/6 - n/3)* 3^(-1 - n/3)*((11*3^(1/3) + 6*2^(1/3)* 3^(2/3))*(3 - Sqrt[2]) + 13*2^(1/6)*(3*Sqrt[2] - 2) + (26*2^(1/6)*(3*Sqrt[2] - 2) - (11*3^(1/3) + 6*2^(1/3)*3^(2/3))* (3 - Sqrt[2]))*Cos[2*n*Pi/3] + 3^(1/6)*(3 - Sqrt[2])*(11*3^(2/3) - 18*2^(1/3))*Sin[2*n*Pi/3])) / (35*(3*Sqrt[2] - 2)), {n, 3, 20}]] (* Vaclav Kotesovec, May 07 2016 *)

Formula

E.g.f.: x^3 * Product_{i=1..3} (i-1)!/(i!-x^i).

Recurrence: 12*a(n) = 12*n*a(n-1) + 6*(n-1)*n*a(n-2) - 4*(n-2)*(n-1)*n*a(n-3) - 2*(n-3)*(n-2)*(n-1)*n*a(n-4) - (n-4)*(n-3)*(n-2)*(n-1)*n*a(n-5) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n*a(n-6). - Vaclav Kotesovec, May 07 2016