A393570 - OEIS

%I #13 Mar 01 2026 22:27:54

%S 1,1,3,7,43,171,1581,8793,108347,774763,11933593,104297733,1927782517,

%T 19911877285,429380343003,5117339645967,126114825517467,

%U 1703436268264475,47228082430915401,712959300038464773,21963252371434460273,366460163952069281073,12417991841852814383103

%N Number of ordered set partitions of [n] with palindromic block sizes.

%H Alois P. Heinz, <a href="/A393570/b393570.txt">Table of n, a(n) for n = 0..442</a>

%F E.g.f.: exp(x)/(1 - Sum_{i>0} x^(2*i)/(i!)^2).

%e The ordered set partition of [5], 25|4|13 has block sizes 2,1,2 so it is counted under a(5) = 171.

%e a(3) = 7: 1|2|3, 1|3|2, 2|1|3, 2|3|1, 3|1|2, 3|2|1, 123.

%p a:= proc(n) option remember; 1+add(a(n-2*j)*

%p       combinat[multinomial](n, n-2*j, j$2), j=1..n/2)

%p     end:

%p seq(a(n), n=0..22);  # _Alois P. Heinz_, Mar 01 2026

%o (PARI) C_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace( exp(x)/(1- sum(i=1,N, x^(2*i)/(i!)^2))))}

%Y Cf. A000670, A016116, A114902.

%K nonn,easy

%O 0,3

%A _John Tyler Rascoe_, Feb 21 2026