A291973 a(n) = (3*n)! * [z^(3*n)] exp(exp(z)/3 + 2*exp(-z/2)*cos(z*sqrt(3)/2)/3 - 1).

1, 1, 11, 365, 25323, 3068521, 583027547, 161601254725, 62042488237755, 31728742163212641, 20963751508027371691, 17461136553331587079965, 17967906090023681913528523, 22459900935806853610377326041, 33617974358392980795259947648187, 59515082206147526028817472280664565

Offset

0,3

Comments

Row sums of A291451.

The number of set partitions of {1,2,...,3n} where the size of every block is a multiple of 3. - Per W. Alexandersson, Jun 20 2024

Links

Andrew Howroyd, Table of n, a(n) for n = 0..100

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(3*n-1,3*k-1) * a(n-k). - Ilya Gutkovskiy, Jan 21 2020

Example

For n=2, there are a(2)=11 partitions of {1,2,...,6} with every block size a multiple of 3: 123456, 123|456, 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235 and 156|234. - Per W. Alexandersson, Jun 20 2024

Maple

A291973 := proc(n) exp(exp(z)/3+2*exp(-z/2)*cos(z*sqrt(3)/2)/3-1):
(3*n)!*coeff(series(%, z, 3*(n+1)), z, 3*n) end:
seq(A291973(n), n=0..15);

Mathematica

P[m_, n_] := P[m, n] = If[n == 0, 1, Sum[Binomial[m*n, m*k]*P[m, n - k]*x, {k, 1, n}]];
a[n_] := Module[{cl = CoefficientList[P[3, n], x]}, Sum[cl[[k + 1]]/k!, {k, 0, n}]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 23 2019, after Peter Luschny in A291451 *)

Prog

(PARI) seq(n)={my(a=vector(n+1)); a[1]=1; for(n=1, n, a[1+n]=sum(k=1, n, binomial(3*n-1, 3*k-1) * a[1+n-k])); a} \\ Andrew Howroyd, Jan 21 2020

Crossrefs

Cf. A291451.

Sequence in context: A257227 A176474 A000464 * A024149 A353934 A333466

Adjacent sequences: A291970 A291971 A291972 * A291974 A291975 A291976

Keyword

nonn

Author

Peter Luschny, Sep 07 2017

Status

approved