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%I #37 Jan 31 2024 17:53:34
%S 1,0,1,1,3,7,21,67,237,907,3741,16507,77517,385627,2024301,11174587,
%T 64673997,391392667,2470864941,16237279867,110858862477,784987938907,
%U 5755734591981,43636725010747,341615028340557,2758165832945947,22940755633301421,196354180631212027
%N Shallow diagonal sums of the triangle k!*Stirling2(n,k): a(n) = Sum_{k=0..floor(n/2)} T(n-k,k) where T is A019538.
%C From _Gus Wiseman_, Jan 08 2019: (Start)
%C Also the number of set partitions of {1,...,n} into blocks of sizes > 1 whose minima form an initial interval of positive integers. For example, the a(5) = 7 set partitions are:
%C {{1,2,3,4,5}}
%C {{1,3},{2,4,5}}
%C {{1,4},{2,3,5}}
%C {{1,5},{2,3,4}}
%C {{1,3,4},{2,5}}
%C {{1,3,5},{2,4}}
%C {{1,4,5},{2,3}}
%C Also the number of ordered set partitions of {1,...,n-k} of length k, for any 0 <= k <= n. For example, the a(5) = 7 ordered set partitions are:
%C {{1,2,3,4}}
%C {{1},{2,3}}
%C {{2},{1,3}}
%C {{3},{1,2}}
%C {{1,2},{3}}
%C {{1,3},{2}}
%C {{2,3},{1}}
%C (End)
%H Alois P. Heinz, <a href="/A105795/b105795.txt">Table of n, a(n) for n = 0..599</a>
%H Giulio Cerbai, <a href="https://arxiv.org/abs/2401.10027">Pattern-avoiding modified ascent sequences</a>, arXiv:2401.10027 [math.CO], 2024. See p. 13.
%F a(n) = sum{k=0..floor(n/2), sum{(-1)^i*binomial(k, i)*(k-i)^(n-k)}}.
%F E.g.f.: Sum_{n>=0} Integral^n (exp(x) - 1)^n dx^n, where integral^n F(x) dx^n is the n-th integration of F(x) with no constant of integration. - _Paul D. Hanna_, Dec 28 2013
%F Formal o.g.f.: 1/(1 + x)*sum {n >= 0} 1/(1 - n*x)*(x/(1 + x))^n = 1 + x^2 + x^3 + 3*x^4 + 7*x^5 + .... Cf. A229046. - _Peter Bala_, Jul 09 2014
%e a(8) = 1!*Stirling2(7,1) + 2!*Stirling2(6,2) + 3!*Stirling2(5,3) + 4!*Stirling2(4,4) = 1 + 62 + 150 + 24 = 237. - _Peter Bala_, Jul 09 2014
%p a:= n-> add(Stirling2(n-k, k)*k!, k=0..n/2):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jul 09 2014
%t Table[Sum[StirlingS2[n-k, k]*k!, {k,0,n/2}],{n,0,20}] (* _Vaclav Kotesovec_, Jul 16 2014 *)
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t Table[Sum[Length[Join@@Permutations/@Select[sps[Range[n-k]],Length[#]==k&]],{k,0,n}],{n,0,10}] (* _Gus Wiseman_, Jan 08 2019 *)
%o (PARI) /* From Paul Barry's formula: */
%o {a(n)=sum(k=0,floor(n/2), sum(i=0,k,(-1)^i*binomial(k, i)*(k-i)^(n-k)))} \\ _Paul D. Hanna_, Dec 28 2013
%o for(n=0,30,print1(a(n),", "))
%o (PARI) /* From e.g.f. series involving iterated integration: */
%o {INTEGRATE(n,F)=local(G=F);for(i=1,n,G=intformal(G));G}
%o {a(n)=local(A=1+x);A=1+sum(k=1,n,INTEGRATE(k,(exp(x+x*O(x^n))-1)^k ));n!*polcoeff(A,n)} \\ _Paul D. Hanna_, Dec 28 2013
%Y Cf. A000110, A000296, A000670, A003101, A008277, A019538, A026898, A287215.
%K easy,nonn
%O 0,5
%A _Paul Barry_, Apr 20 2005
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