A059099 Expansion of exp(exp(x)-1)/(2-exp(x)).

1, 2, 7, 33, 198, 1453, 12669, 128320, 1482721, 19260421, 277913552, 4410640919, 76360030701, 1432144732762, 28926138244883, 625974400305541, 14449445989893990, 354384475357492593, 9202837263156670345, 252260867710562944224, 7278710072406887897461

Offset: 0

Author

Vladeta Jovovic, Jan 02 2001

Keywords

• easy
• nonn

Comments

Row sums of A227343. - Peter Bala, Jul 11 2013

The sequence gives the number of barred preferential arrangements of an n-set having one bar, where one fixed section is a free section and elements which are to go into the other section are partitioned into unordered nonempty subsets. - Sithembele Nkonkobe, Jul 02 2015

Examples

			exp(exp(x)-1)/(2-exp(x)) = 1 + 2*x + 7/2*x^2 + 11/2*x^3 + 33/4*x^4 + 1453/120*x^5 + 4223/240*x^6 + 1604/63*x^7 + ...
		

Links

• Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See pp. 12, 19, 29.
• S. Nkonkobe and V. Murali, A study of a family of generating functions of Nelsen-Schmidt type and some identities on restricted barred preferential arrangements, arXiv:1503.06172 [math.CO], 2015.

Crossrefs

Row sums of A059098.

Cf. A001861, A000110, A005493, A049020, A227343.

Programs

Maple

s := series(exp(exp(x)-1)/(2-exp(x)), x, 60): for i from 0 to 50 do printf(`%d,`,i!*coeff(s,x,i)) od:

Mathematica

CoefficientList[Series[E^(E^x-1)/(2-E^x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jul 02 2015 *)

Formula

a(n) = Sum_{m=0..n} Sum_{i=0..n} Stirling2(n, i)*Product_{j=1..m} (i-j+1).

Stirling transform of A000522. - Vladeta Jovovic, May 10 2004

a(n) ~ n!*exp(1)/(2*(log(2))^(n+1)). - Vaclav Kotesovec, Jul 02 2015

Extensions

More terms from James Sellers, Jan 03 2001