A102286 Total number of odd blocks in all partitions of n-set.

1, 2, 7, 24, 96, 418, 1989, 10216, 56275, 330424, 2057672, 13532060, 93633021, 679473694, 5156626991, 40824399712, 336406367196, 2879570703510, 25557841113625, 234822774979908, 2230107923204443, 21861817965483016, 220940261740238140, 2299258336094622008

Offset

1,2

Comments

a(n) is also the number of set partitions of {1,2,...,n+1} in which the element 1 is in an even size block. - Geoffrey Critzer, Apr 02 2013

Links

Alois P. Heinz, Table of n, a(n) for n = 1..575

Formula

E.g.f: sinh(x)*exp(exp(x)-1).

a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1) * Bell(n-2*k-1). - Ilya Gutkovskiy, Apr 10 2022

Example

a(3)=7 because we have (123), (1)/23, 12/(3), 13/(2), (1)/(2)/(3); the odd blocks are shown between parentheses.

Maple

G:=sinh(x)*exp(exp(x)-1): Gser:=series(G, x=0, 30): seq(n!*coeff(Gser, x^n), n=1..25); # Emeric Deutsch
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
       add((p->(p+[0, `if`(i::odd, j, 0)*p[1]]))(
       b(n-i*j, i-1))*multinomial(n, n-i*j, i$j)/j!, j=0..n/i))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 16 2015

Mathematica

Range[0, nn]! CoefficientList[
  D[Series[Exp[ (Cosh[x] - 1) + y Sinh[x]], {x, 0, nn}], y] /. y -> 1, x] (* Geoffrey Critzer, Aug 28 2012 *)
With[{nn=30}, CoefficientList[Series[Sinh[x]Exp[Exp[x]-1], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jul 03 2021 *)

Crossrefs

Cf. A000110, A000296, A005493.

Sequence in context: A150423 A150424 A150425 * A150426 A150427 A150428

Adjacent sequences: A102283 A102284 A102285 * A102287 A102288 A102289

Keyword

easy,nonn

Author

Vladeta Jovovic, Feb 19 2005

Extensions

More terms from Emeric Deutsch, Mar 04 2005

Status

approved