A105795 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.

1, 0, 1, 1, 3, 7, 21, 67, 237, 907, 3741, 16507, 77517, 385627, 2024301, 11174587, 64673997, 391392667, 2470864941, 16237279867, 110858862477, 784987938907, 5755734591981, 43636725010747, 341615028340557, 2758165832945947, 22940755633301421, 196354180631212027

Offset: 0

Author

Paul Barry, Apr 20 2005

Keywords

• easy
• nonn

Comments

From Gus Wiseman, Jan 08 2019: (Start)

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:

{{1,2,3,4,5}}

{{1,3},{2,4,5}}

{{1,4},{2,3,5}}

{{1,5},{2,3,4}}

{{1,3,4},{2,5}}

{{1,3,5},{2,4}}

{{1,4,5},{2,3}}

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:

{{1,2,3,4}}

{{1},{2,3}}

{{2},{1,3}}

{{3},{1,2}}

{{1,2},{3}}

{{1,3},{2}}

{{2,3},{1}}

(End)

Examples

			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
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..599
• Giulio Cerbai, Pattern-avoiding modified ascent sequences, arXiv:2401.10027 [math.CO], 2024. See p. 13.

Crossrefs

Cf. A000110, A000296, A000670, A003101, A008277, A019538, A026898, A287215.

Programs

Maple

a:= n-> add(Stirling2(n-k, k)*k!, k=0..n/2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 09 2014

Mathematica

Table[Sum[StirlingS2[n-k, k]*k!, {k,0,n/2}],{n,0,20}] (* Vaclav Kotesovec, Jul 16 2014 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Length[Join@@Permutations/@Select[sps[Range[n-k]],Length[#]==k&]],{k,0,n}],{n,0,10}] (* Gus Wiseman, Jan 08 2019 *)

PARI

/* From Paul Barry's formula: */
{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
for(n=0,30,print1(a(n),", "))

PARI

/* From e.g.f. series involving iterated integration: */
{INTEGRATE(n,F)=local(G=F);for(i=1,n,G=intformal(G));G}
{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

Formula

a(n) = sum{k=0..floor(n/2), sum{(-1)^i*binomial(k, i)*(k-i)^(n-k)}}.

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

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