A351932 Number of set partitions of [n] such that block sizes are either 1 or 4.

1, 1, 1, 1, 2, 6, 16, 36, 106, 442, 1786, 6106, 23596, 120836, 631632, 2854216, 13590396, 81258556, 510768316, 2839808572, 16008902296, 108643656136, 787965516416, 5161270717296, 33513036683512, 253407796702776, 2065728484459576, 15485032349429176, 113510664648701776

Offset: 0

Author

Seiichi Manyama, Feb 26 2022

Keywords

• nonn

Crossrefs

Cf. A000085, A190865, A275423, A275425.

Cf. A190452, A190875, A351930.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1,
     `if`(n<4, 0, a(n-4)*binomial(n-1, 3))+a(n-1))
    end:
seq(a(n), n=0..28);  # Alois P. Heinz, Feb 26 2022
seq(round(evalf(hypergeom([-n/4,(1-n)/4,(2-n)/4,(3-n)/4],[],32/3))),n=0..28);  # Karol A. Penson, Jul 28 2023

PARI

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x+x^4/4!)))

PARI

a(n) = n!*sum(k=0, n\4, 1/4!^k*binomial(n-3*k, k)/(n-3*k)!);

PARI

a(n) = if(n<4, 1, a(n-1)+binomial(n-1, 3)*a(n-4));

Formula

E.g.f.: exp(x + x^4/24).

a(n) = n! * Sum_{k=0..floor(n/4)} (1/24)^k * binomial(n-3*k,k)/(n-3*k)!.

a(n) = a(n-1) + binomial(n-1,3) * a(n-4) for n > 3.

a(n) = hypergeom([-n/4,(1-n)/4,(2-n)/4,(3-n)/4],[],32/3), Karol A. Penson, Jul 28 2023.