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URL: https://oeis.org/A392810

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A392810
Number of ordered set partitions of [n] whose block sizes are a palindromic sequence with no adjacent equal elements.
1
1, 1, 1, 1, 13, 51, 31, 1653, 6077, 41779, 638701, 5158473, 15616855, 689190373, 4459154883, 59035926771, 1195876420317, 15970500617363, 99735648719485, 4555440147330105, 42624771030616373, 896478845515669809, 21027550319473343163, 399975004886038739435
OFFSET
0,5
LINKS
FORMULA
E.g.f.: 1 + Sum_{i>0} (x^i * (1 - b(i,x)))/(i! * (1 - Sum_{k>0} b(k,x))) where b(i,x) = 1/(1 + (i!)^2 * x^(-2*i)).
EXAMPLE
The ordered set partition of [5], 2|345|1 has block sizes 1,3,1 so it is counted under a(5) = 51.
a(4) = 13: 1234, 1|23|4, 1|24|3, 1|34|2, 2|13|4, 2|14|3, 2|34|1, 3|12|4, 3|14|2, 3|24|1, 4|12|3, 4|13|2, 4|23|1.
MAPLE
b:= proc(n, i) option remember; `if`(i=n, 0, 1)+add(`if`(i=j, 0,
b(n-2*j, j)*combinat[multinomial](n, n-2*j, j$2)), j=1..(n-1)/2)
end:
a:= n-> b(n, -1):
seq(a(n), n=0..23); # Alois P. Heinz, Mar 02 2026
PROG
(PARI)
b(f) = {1/(1 + (f!)^2 * x^(-2*f))}
D_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(1 + sum(i=1, N, (x^i * (1 - b(i)))/((i!)*(1 - sum(k=1, N-i, b(k)))))))}
CROSSREFS
KEYWORD
nonn
AUTHOR
John Tyler Rascoe, Feb 28 2026
STATUS
approved