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A351932
Number of set partitions of [n] such that block sizes are either 1 or 4.
2
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
0,5
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.
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
PROG
(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));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Feb 26 2022
STATUS
approved