%I #32 May 12 2024 11:29:10

%S 1,0,0,0,1,5,21,91,422,2103,11226,63879,385691,2461004,16535820,

%T 116628147,861033654,6637143698,53297137552,444940442553,

%U 3854539901147,34592812084693,321125878230123,3079144039478532,30457076370822777,310407099470429818,3255972198123974137,35114803641531204063

%N Number of partitions of the set {1,2,...,n} such that no block is a sequence of consecutive integers (including 1-element blocks)

%C Some similar results appear in Klazar (see links).

%D Richard Stanley, Enumerative Combinatorics, volume 1, second edition, Cambridge Univ Press, 2011, page 192, solution 111.

%H Alois P. Heinz, <a href="/A168444/b168444.txt">Table of n, a(n) for n = 0..500</a>

%H Martin Klazar, <a href="http://dx.doi.org/10.1016/S0097-3165(03)00014-1">Bell numbers, their relatives and algebraic differential equations</a>, J. Combin. Theory, A 102 (2003), 63-87.

%F Ordinary g.f.: (1-x)F(x(1-x)), where F(x) = sum_{n>=0} B(n)x^n (the ordinary g.f. for the Bell numbers)

%F a(n) = b(n)-b(n-1), b(n) = if n=0 then 1 else sum(binomial(k,n-k)*(-1)^(n-k)*B(k),k=ceiling(n/2)..n). - _Vladimir Kruchinin_, Sep 09 2010

%e For n=5 the a(5) = 5 partitions are 13-245, 14-235, 24-135, 25-135, 35-124.

%p with(combinat): y:=sum(bell(n)*x^n,n=0..20): z:=(1-x)*subs(x=x*(1-x),y): taylor(z,x=0,21);

%t nn = 20; b := Sum[BellB[n] (x - x^2)^n, {n, 0, nn}]; CoefficientList[ Series[ (1-x) b, {x, 0, nn}], x] (* _Geoffrey Critzer_, Jun 01 2013 *)

%o (Maxima) b(n):=if n=0 then 1 else sum(binomial(k,n-k)*(-1)^(n-k)*belln(k),k,ceiling(n/2),n); a(n):=if n=0 then 1 else b(n)-b(n-1); /* _Vladimir Kruchinin_, Sep 09 2010 */

%o (PARI)

%o N=66; x = 'x+O('x^N);

%o B = serlaplace(exp(exp(x)-1));

%o gf = (1-x)*subst(B,'x, x*(1-x));

%o Vec(gf) \\ _Joerg Arndt_, Jun 01 2013

%o (Magma)

%o b:= func< n | n eq 0 select 1 else (&+[(-1)^(n+j)*Binomial(j,n-j)*Bell(j): j in [Ceiling(n/2)..n]]) >;

%o A168444:= func< n | n eq 0 select 1 else b(n)-b(n-1) >;

%o [A168444(n): n in [0..30]]; // _G. C. Greubel_, May 12 2024

%o (SageMath)

%o @CachedFunction

%o def b(n): return 1 if (n==0) else sum((-1)^(n+j)*binomial(j,n-j)*bell_number(j) for j in range((n//2), n+1))

%o def A168444(n): return 1 if (n==0) else b(n) - b(n-1)

%o [A168444(n) for n in range(31)] # _G. C. Greubel_, May 12 2024

%Y Column k=0 of A177254.

%K easy,nonn

%O 0,6

%A _Richard Stanley_, Nov 25 2009

%E Added more terms, _Joerg Arndt_, Jun 01 2013