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multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[EvenQ[i] && OddQ[j], 0, b[n-i*j, i- 1] * multinomial[n, Join[{n - i*j}, Array[i &, j]]]/j!*i!^j], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)
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Alois P. Heinz, <a href="/A102760/b102760.txt">Table of n, a(n) for n = 0..445</a>
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(`if`(i::even and j::odd, 0, b(n-i*j, i-1)*
multinomial(n, n-i*j, i$j)/j!*i!^j), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..25); # Alois P. Heinz, May 10 2016
1, 1, 1, 7, 37, 241, 1381, 13231, 140617, 1483777, 16211881, 217551511, 3384215341, 50221272817, 782154787597, 13913712591871, 272739557719441, 5282625708305281, 106588332600443857, 2354480141600267047, 56238135934525073461, 1338131691952924913521
1,3
0,4
a(0)=1 prepended by Alois P. Heinz, May 10 2016
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editing
_Vladeta Jovovic (vladeta(AT)eunet.rs), _, Feb 10 2005