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A069321 Stirling transform of A001563: a(0)=1, a(n) = Sum(stirling2(n,k)*k*k!,k=1..n) for n=1,2... 11
1, 1, 5, 31, 233, 2071, 21305, 249271, 3270713, 47580151, 760192505, 13234467511, 249383390393, 5057242311031, 109820924003705, 2542685745501751, 62527556173577273, 1627581948113854711, 44708026328035782905, 1292443104462527895991, 39223568601129844839353 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..200

B. Cloitre, On the fractal behavior of primes, 2011. [Dead link]

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.

FORMULA

Representation as an infinite series, in Maple notation: a(0)=1, a(n) = Sum_{k>=2}(k^n*(k-1)/(2^k))/4, n=1, 2, ... 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

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-Fran├žois Alcover, Mar 30 2016 *)

PROG

(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

CROSSREFS

Cf. A001563.

Cf. A005649, A000670.

Sequence in context: A052773 A062147 A213048 * A211179 A177797 A293717

Adjacent sequences:  A069318 A069319 A069320 * A069322 A069323 A069324

KEYWORD

nonn

AUTHOR

Karol A. Penson, Mar 14 2002

STATUS

approved

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Last modified February 18 02:57 EST 2020. Contains 332006 sequences. (Running on oeis4.)