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%I #15 Jan 09 2024 18:30:35
%S 0,0,1,9,73,637,6220,68414,844067,11589987,175612351,2912695193,
%T 52502754076,1022091626496,21372127906257,477737240288353,
%U 11368449905784189,286935157928114989,7656210527253978232
%N Number of permutations of length n containing exactly one occurrence of the pattern 1-32.
%H Paolo Xausa, <a href="/A086226/b086226.txt">Table of n, a(n) for n = 0..400</a>
%H A. Claesson and T. Mansour, <a href="https://doi.org/10.1016/S0196-8858(02)00012-X">Counting Occurrences of a Pattern of Type (1,2) or (2,1) in Permutations</a>, Advances in Applied Mathematics, 29 (2002), 293-310.
%F a(0)=0; a(n)=a(n-1)+sum(k=1, n-1, binomial(n, k)*a(k)+binomial(n-1, k-1)*B(k))) where B(k) is the k-th Bell number.
%t A086226[n_] := A086226[n] = If[n==0,0,A086226[n-1] + Sum[Binomial[n,k] A086226[k] + Binomial[n-1,k-1] BellB[k],{k,n-1}]];
%t Array[A086226,25,0] (* _Paolo Xausa_, Jan 09 2024 *)
%o (PARI) B(n)=round(exp(-1)*sum(k=0,200,k^n/k!));
%o an=vector(100); a(n)=if(n<1,0,an[n]);
%o for(n=1,30,an[n]=a(n-1)+sum(k=1,n-1,binomial(n,k)*a(k)+binomial(n-1,k-1)*B(k)));
%o an
%Y Cf. A000110 (Bell numbers).
%K nonn
%O 0,4
%A _Benoit Cloitre_, Aug 28 2003
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