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b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i > n, 0, b[n, i + 1, If[i == k, 0, k]] + If[i == k, 0, b[n - i, i, k] Binomial[n, i]]]];
a[n_] := b[n, 1, 0] - b[n, 1, 2];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)
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a(n) ~ c * n!, where c = A247551/2 = 1.26473873603957632409005807712697712... - Vaclav Kotesovec, Sep 28 2019
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Alois P. Heinz, <a href="/A327828/b327828.txt">Table of n, a(n) for n = 0..450</a>
a
Sum of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts incorporating 2.
0, 0, 1, 3, 18, 100, 705, 5166, 44856, 413316, 4297635, 47906650, 586050828, 7669704978, 108433645502, 1632017808435, 26240224612920, 446861879976600, 8063224431751719, 153335328111105282, 3070484092409318100, 64508501542986638550, 1420061287311444508962
0,14
b:= proc(n, i, k) option remember; `if`(n=0, 1,
`if`(i>n, 0, b(n, i+1, `if`(i=k, 0, k))+
`if`(i=k, 0, b(n-i, i, k)*binomial(n, i))))
end:
a:= n-> b(n, 1, 0)-b(n, 1, 2):
seq(a(n), n=0..23);
allocated for Alois P. Heinz
a
18, 100, 705, 5166
0,1
Column k=2 of A327801.
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Alois P. Heinz, Sep 26 2019
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