

A214406


Triangle of secondorder 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
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OFFSET

0,5


COMMENTS

The secondorder 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*n1, if x_j < x_(j+1). We define T(n,k), the secondorder 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, Secondorder 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], 20052006.


FORMULA

T(n,k) = sum {i = 0..k} (1)^(ik)*binomial(2*n+1,ki)*S(n+i,i), where S(n,k) = 1/(2^k*k!)*sum {j = 0..k} (1)^(kj)* binomial(k,j) *(2*j+1)^n = A039755(n,k).
It appears that sum {k = 0..n} (1)^(k+1)*T(n,k)/((2*nk)*binomial(2*n,k)) = (1)^n *(2^n2)*Bernoulli(n)/n.
Recurrence equation: T(n,k) = (4*n2*k1)*T(n1,k1) + (2*k+1)*T(n1,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) = (1x^2)^(2*n)* d/dx[x/(1x^2)^(2*n1)*R(n1,x^2)]. Equivalently, x*R(n,x^2)/(1x^2)^(2*n+1)) = D^n(x), where D is the differential operator x/(1x^2)*d/dx.
Another recurrence is R(n+1,x) = 2*x*(1x)*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)/(1x)^(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 semiorthogonality property int {x = 0..inf} (1x)*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)/(1x)^(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] = (4n2k1) T[n1, k1] + (2k+1) T[n1, k]; T[_, 0] = 1; T[_, _] = 0;
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* JeanFrançois 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 JeanFrançois Alcover, Nov 11 2019


STATUS

approved



