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A214406 Triangle of second-order Eulerian numbers of type B. 6
1, 1, 1, 1, 8, 3, 1, 33, 71, 15, 1, 112, 718, 744, 105, 1, 353, 5270, 14542, 9129, 945, 1, 1080, 33057, 191384, 300291, 129072, 10395, 1, 3265, 190125, 2033885, 6338915, 6524739, 2071215, 135135, 1, 9824, 1038780, 18990320, 103829590, 204889344, 150895836, 37237680, 2027025 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The second-order Eulerian numbers A008517 count Stirling permutations by ascents. A Stirling permutation of order n is a permutation of the multiset {1,1,2,2,...,n,n} such that for each i, 1 <= i <= n, the elements lying between the two occurrences of i are larger than i.

We define a signed Stirling permutation of order n to be a vector (x_0,x_1,...,x_(2*n)) such that x_0 = 0 and (|x_1|,...,|x_(2*n)|) is a Stirling permutation of order n. We say that a signed Stirling permutation (x_0,x_1,...,x_(2*n)) has an ascent at position j, 0 <= j <= 2*n-1, if |x_j| < |x_(j+1)|. We define T(n,k), the second-order Eulerian numbers of type B, as the number of signed Stirling permutations of order n having k ascents. An example is given below.

LINKS

Table of n, a(n) for n=0..44.

P. Bala, Second-order Eulerian numbers of type B

P. Bala, Notes on A214406

L. Liu, Y. Wang, A unified approach to polynomial sequences with only real zeros arXiv:math/0509207 [math.CO], 2005-2006.

FORMULA

T(n,k) = sum {i = 0..k} (-1)^(i-k)*binomial(2*n+1,k-i)*S(n+i,i), where S(n,k) = 1/(2^k*k!)*sum {j = 0..k} (-1)^(k-j)* binomial(k,j) *(2*j+1)^n = A039755(n,k).

It appears that sum {k = 0..n} (-1)^(k+1)*T(n,k)/((2*n-k)*binomial(2*n,k)) = (-1)^n *(2^n-2)*Bernoulli(n)/n.

Recurrence equation: T(n,k) = (4*n-2*k-1)*T(n-1,k-1) + (2*k+1)*T(n-1,k), for n,k >= 0.

The row polynomials R(n,x) may be calculated by means of the recurrence equation R(0,x) = 1 and for n >=0, R(n,x^2) = (1-x^2)^(2*n)* d/dx[x/(1-x^2)^(2*n-1)*R(n-1,x^2)]. Equivalently, x*R(n,x^2)/(1-x^2)^(2*n+1)) = D^n(x), where D is the differential operator x/(1-x^2)*d/dx.

Another recurrence is R(n+1,x) = 2*x*(1-x)*d/dx(R(n,x)) + (1+(4*n+1)*x)*R(n,x). It follows that the row polynomials R(n,x) have only real zeros (apply Liu and Wang, Corollary 1.2 with f(x) = R(n,x) and g(x) = R'(n,x)).

For n >= 0, the rational functions Q(n,x) := R(n,x)/(1-x)^(2*n+1) are the o.g.f.'s for the diagonals of the type B Stirling numbers of the second kind A039755. They appear to satisfy the semi-orthogonality property int {x = 0..inf} (1-x)*Q(n,x)*Q(m,x) dx = (-1)^n*(2^(n+m)-2)*Bernoulli(n+m)/(n+m), for n,m >= 0 but excluding the case (n,m) = (0,0). A similar result holds for the row polynomials of A185896.

Row sums are A001813.

Define functions F(n,z) := sum {k >= 0} (2*k+1)^(k+n)*z^k/k!, n = 0,1,2,.... Then exp(-x/2)*F(n,x/2*exp(-x)) = R(n,x)/(1-x)^(2*n+1). - Peter Bala, Jul 26 2012

EXAMPLE

Row 2: [1,8,3]:

Signed Stirling permutations of order 2

= = = = = = = = = = = = = = = = = = = =

..............ascents...................ascents

(0 2 2 1 1)......1.......(0 -2 -2 1 1).....1

(0 1 2 2 1)......2.......(0 1 -2 -2 1).....2

(0 1 1 2 2)......2.......(0 1 1 -2 -2).....1

(0 2 2 -1 -1)....1.......(0 -2 -2 -1 -1)...1

(0 -1 2 2 -1)....1.......(0 -1 -2 -2 -1)...1

(0 -1 -1 2 2)....1.......(0 -1 -1 -2 -2)...0

............................................

Triangle begins

.n\k.|..0.....1......2.......3......4........5......6

= = = = = = = = = = = = = = = = = = = = = = = = = = =

..0..|..1

..1..|..1.....1

..2..|..1.....8......3

..3..|..1....33.....71......15

..4..|..1...112....718.....744....105

..5..|..1...353...5270...14542...9129......945

..6..|..1..1080..33057..191384..300291..129072..10395

...

Recurrence example: T(4,2) = 11*T(3,1) + 5*T(3,2) = 11*33 + 5*71 = 718.

MATHEMATICA

T[n_, k_] /; 0 < k <= n := T[n, k] = (4n-2k-1) T[n-1, k-1] + (2k+1) T[n-1, k]; T[_, 0] = 1; T[_, _] = 0;

Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 11 2019 *)

CROSSREFS

Cf. A001813 (row sums), A008517, A039755, A185896, A034940.

Sequence in context: A056030 A195473 A195728 * A176454 A199380 A104842

Adjacent sequences:  A214403 A214404 A214405 * A214407 A214408 A214409

KEYWORD

nonn,easy,tabl

AUTHOR

Peter Bala, Jul 17 2012

EXTENSIONS

Missing 1 in data inserted by Jean-François Alcover, Nov 11 2019

STATUS

approved

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Last modified October 1 04:06 EDT 2020. Contains 337441 sequences. (Running on oeis4.)