A227343 -id:A227343 - cp's OEIS frontend

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

Vladeta Jovovic, Jan 02 2001

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

			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 + ...

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.

Row sums of A059098.

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

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:

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

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

More terms from James Sellers, Jan 03 2001

A227342 Expansion of (1 - t)*(1 + t)^x.

Original entry on oeis.org

1, -1, 1, 0, -3, 1, 0, 5, -6, 1, 0, -14, 23, -10, 1, 0, 54, -105, 65, -15, 1, 0, -264, 574, -435, 145, -21, 1, 0, 1560, -3682, 3199, -1330, 280, -28, 1, 0, -10800, 27180, -26124, 12649, -3360, 490, -36, 1, 0, 85680, -227196, 236312, -128205, 40089, -7434, 798, -45, 1

Offset: 0

Peter Bala, Jul 11 2013

Cf. A105793. Inverse array is A227343.
The e.g.f. has the form A(t)*exp(x*B(t)), where A(t) = 1 - t and B(t) = log(1 + t). Thus the row polynomials of this triangle form a Sheffer sequence for the pair (2 - exp(t), exp(t) - 1) (see Roman, p. 17).
Let x_(k) := x*(x-1)*...*(x-k+1) denote the k-th falling factorial polynomial. Define a sequence x_[n] of basis polynomials for the polynomial algebra C[x] by setting x_[0] = 1, and setting x_[n] = x_(n-1)*(x - 2*n + 1) for n >= 1. The sequence begins [1, x-1, x*(x-3), x*(x-1)*(x-5), x*(x-1)*(x-2)*(x-7), ...]. Then this is the triangle of connection constants for expressing the basis polynomials x_[n] as a linear combination of the monomial polynomials x^k, that is, x_[n] = Sum_{k = 0..n} T(n,k) x^k. An example is given below.

			Triangle begins:
n\k|  0       1        2       3        4      5      6    7    8   9
===|==================================================================
0  |  1
1  | -1,      1;
2  |  0,     -3,       1;
3  |  0,      5,      -6,      1;
4  |  0,    -14,      23,    -10,       1;
5  |  0,     54,    -105,     65,     -15,     1;
6  |  0,   -264,     574,   -435,     145,   -21,     1;
7  |  0,   1560,   -3682,   3199,   -1330,   280,   -28,   1;
8  |  0, -10800,   27180, -26124,   12649, -3360,   490, -36,   1;
9  |  0,  85680, -227196, 236312, -128205, 40089, -7434, 798, -45,  1;
...
Connection constants. Row 4 = [0, -14 ,23, -10, 1]:
-14*x + 23*x^2 - 10*x^3 + x^4 = x*(x-1)*(x-2)*(x-7) = x_[4].

S. Roman, The umbral calculus, Pure and Applied Mathematics 111, Academic Press Inc., New York, 1984. Reprinted by Dover in 2005.

Eric Weisstein's World of Mathematics, Sheffer Sequence

Cf. A048994, A105793, A227343 (matrix inverse).

T(n,k) = Stirling1(n,k) - n*Stirling1(n-1,k).
E.g.f.: (1 - t)*(1 + t)^x = 1 + (-1 + x)*t + (-3*x + x^2)*t^2/2! + (5*x - 6*x^2 + x^3)*t^3/3! + ....
E.g.f. for column k: (1/k!)*(1 - t)*(log(1 + t))^k.
The row polynomials R(n,x) satisfy the Sheffer identity R(n,x + y) = Sum_{k = 0..n} binomial(n,k)*y_(k)*R(n-k,x), where y_(k) is the falling factorial. As a particular case we have the identity R(n,x + 1) - R(n,x) = n*R(n-1,x) for n >= 1.

cp's OEIS Frontend

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

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