A069321 Stirling transform of A001563: a(0) = 1 and a(n) = Sum_{k=1..n} Stirling2(n,k)*k*k! for n >= 1.

1, 1, 5, 31, 233, 2071, 21305, 249271, 3270713, 47580151, 760192505, 13234467511, 249383390393, 5057242311031, 109820924003705, 2542685745501751, 62527556173577273, 1627581948113854711, 44708026328035782905, 1292443104462527895991, 39223568601129844839353

Offset: 0

Author

Karol A. Penson, Mar 14 2002

Keywords

• nonn

Comments

The number of compatible bipartitions of a set of cardinality n for which at least one subset is not underlined. E.g., for n=2 there are 5 such bipartitions: {1 2}, {1}{2}, {2}{1}, {1}{2}, {2}{1}. A005649 is the number of bipartitions of a set of cardinality n. A000670 is the number of bipartitions of a set of cardinality n with none of the subsets underlined. - Kyle Petersen, Mar 31 2005

a(n) is the cardinality of the image set summed over "all surjections". All surjections means: onto functions f:{1, 2, ..., n} -> {1, 2, ..., k} for every k, 1 <= k <= n. a(n) = Sum_{k=1..n} A019538(n, k)*k. - Geoffrey Critzer, Nov 12 2012

From Gus Wiseman, Jan 15 2022: (Start)

For n > 1, also the number of finite sequences of length n + 1 covering an initial interval of positive integers with at least two adjacent equal parts, or non-anti-run patterns, ranked by the intersection of A348612 and A333217. The complement is counted by A005649. For example, the a(3) = 31 patterns, grouped by sum, are:

(1111) (1222) (1122) (1112) (1233) (1223)

(2122) (1221) (1121) (1332) (1322)

(2212) (2112) (1211) (2133) (2213)

(2221) (2211) (2111) (2331) (2231)

(1123) (3312) (3122)

(1132) (3321) (3221)

(2113)

(2311)

(3112)

(3211)

Also the number of ordered set partitions of {1,...,n + 1} with two successive vertices together in some block.

(End)

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• Benoit Cloitre, On the fractal behavior of primes, 2011. [internet archive]
• Benoit Cloitre, On the fractal behavior of primes, 2011.
• D. Foata and D. Zeilberger, The Graphical Major Index, arXiv:math/9406220 [math.CO], 1994.
• D. Foata and D. Zeilberger, Graphical major indices, J. Comput. Appl. Math. 68 (1996), no. 1-2, 79-101.

Crossrefs

Cf. A000629, A001563, A232472.

The complement is counted by A005649.

A version for permutations of prime indices is A336107.

A version for factorizations is A348616.

Dominated (n > 1) by A350252, complement A345194, compositions A345192.

A000670 = patterns, ranked by A333217.

A001250 = alternating permutations, complement A348615.

A003242 = anti-run compositions, ranked by A333489.

A019536 = necklace patterns.

A226316 = patterns avoiding (1,2,3), weakly A052709, complement A335515.

A261983 = not-anti-run compositions, ranked by A348612.

A333381 = anti-runs of standard compositions.

Cf. A000110, A008277, A055932, A106356, A238424, A333382, A335456, A336103, A349050, A349055, A350138.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n, j), j=1..n))
    end:
a:= n-> `if`(n=0, 2, b(n+1)-b(n))/2:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 02 2018

Mathematica

max = 20; t = Sum[n^(n - 1)x^n/n!, {n, 1, max}]; Range[0, max]!CoefficientList[Series[D[1/(1 - y(Exp[x] - 1)), y] /. y -> 1, {x, 0, max}], x] (* Geoffrey Critzer, Nov 12 2012 *)
Prepend[Table[Sum[StirlingS2[n, k]*k*k!, {k, n}], {n, 18}], 1] (* Michael De Vlieger, Jan 03 2016 *)
a[n_] := (PolyLog[-n-1, 1/2] - PolyLog[-n, 1/2])/4; a[0] = 1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 30 2016 *)
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],MemberQ[Differences[#],0]&]],{n,0,8}] (* Gus Wiseman, Jan 15 2022 *)

PARI

{a(n)=polcoeff(1+sum(m=1, n, (2*m-1)!/(m-1)!*x^m/prod(k=1, m, 1+(m+k-1)*x+x*O(x^n))), n)} \\ Paul D. Hanna, Oct 28 2013

Formula

Representation as an infinite series: a(0) = 1 and a(n) = Sum_{k>=2} (k^n*(k-1)/(2^k))/4 for n >= 1. This is a Dobinski-type summation formula.

E.g.f.: (exp(x) - 1)/((2 - exp(x))^2).

a(n) = (1/2)*(A000670(n+1) - A000670(n)).

O.g.f.: 1 + Sum_{n >= 1} (2*n-1)!/(n-1)! * x^n / (Product_{k=1..n} (1 + (n + k - 1)*x)). - Paul D. Hanna, Oct 28 2013

a(n) = (A000629(n+1) - A000629(n))/4. - Benoit Cloitre, Oct 20 2002

a(n) = A232472(n-1)/2. - Vincenzo Librandi, Jan 03 2016

a(n) ~ n! * n / (4 * (log(2))^(n+2)). - Vaclav Kotesovec, Jul 01 2018

a(n > 0) = A000607(n + 1) - A005649(n). - Gus Wiseman, Jan 15 2022