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A131689 Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows (n >= 0, 0 <= k <= n). 35
1, 0, 1, 0, 1, 2, 0, 1, 6, 6, 0, 1, 14, 36, 24, 0, 1, 30, 150, 240, 120, 0, 1, 62, 540, 1560, 1800, 720, 0, 1, 126, 1806, 8400, 16800, 15120, 5040, 0, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320, 0, 1, 510, 18150, 186480, 834120, 1905120, 2328480 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,1,0,2,0,3,0,4,0,5,0,6,0,7,0,...] DELTA [1,1,2,2,3,3,4,4,5,5,6,6,...] where DELTA is the operator defined in A084938; another version of A019538.

See also A019538: version with n > 0 and k > 0. - Philippe Deléham, Nov 03 2008

From Peter Bala, Jul 21 2014: (Start)

T(n,k) gives the number of (k-1)-dimensional faces in the interior of the first barycentric subdivision of the standard (n-1)-dimensional simplex. For example, the barycentric subdivision of the 1-simplex is o--o--o, with 1 interior vertex and 2 interior edges, giving T(2,1) = 1 and T(2,2) = 2.

This triangle is used when calculating the face vectors of the barycentric subdivision of a simplicial complex. Let S be an n-dimensional simplicial complex and write f_k for the number of k-dimensional faces of S, with the usual convention that f_(-1) = 1, so that F := (f_(-1), f_0, f_1,...,f_n) is the f-vector of S. If M(n) denotes the square matrix formed from the first n+1 rows and n+1 columns of the present triangle, then the vector F*M(n) is the f-vector of the first barycentric subdivision of the simplicial complex S (Brenti and Welker, Lemma 2.1). For example, the rows of Pascal's triangle A007318 (but with row and column indexing starting at -1) are the f-vectors for the standard n-simplexes. It follows that A007318*A131689, which equals A028246, is the array of f-vectors of the first barycentric subdivision of standard n-simplexes. (End)

This triangle T(n, k) appears in the o.g.f. G(n, x) = Sum_{m>=0} S(n, m)*x^m with S(n, m) = Sum_{j=0..m} j^n for n >= 1 as G(n, x) = Sum_{k=1..n} (x^k/(1 - x)^(k+2))*T(n, k). See also the Eulerian triangle A008292 with a Mar 31 2017 comment for a rewritten form. For the e.g.f see A028246 with a Mar 13 2017 comment. - Wolfdieter Lang, Mar 31 2017

LINKS

Vincenzo Librandi, Rows n = 0..100, flattened

F. Brenti and V. Welker, f-vectors of barycentric subdivisions, arXiv:math/0606356 [math.CO], Math. Z., 259(4), 849-865, 2008.

Germain Kreweras, Une dualité élémentaire souvent utile dans les problèmes combinatoires, Mathématiques et Sciences Humaines 3 (1963): 31-41.

Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.

M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq. 14 (2011) # 11.9.7.

Wikipedia, Barycentric subdivision

Wikipedia, Simplicial complex

Wikipedia, Simplex

FORMULA

T(n,k) = k*(T(n-1,k-1) + T(n-1,k)) with T(0,0)=1. Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A000629(n), A033999(n), A000007(n), A000670(n), A004123(n+1), A032033(n), A094417(n), A094418(n), A094419(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6 respectively.

Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000142(n), A000670(n), A122704(n) for x=-1, 0, 1, 2 respectively. - Philippe Deléham, Oct 09 2007

Sum_{k=0..n} (-1)^k*T(n,k)/(k+1) = Bernoulli numbers A027641(n)/A027642(n). - Peter Luschny, Sep 17 2011

G.f.: F(x,t) = 1 + x*t + (x+x^2)*t^2/2! + (x+6*x^2+6*x^3)*t^3/3! + ... = Sum_{n>=0} R(n,x)*t^n/n!.

The row polynomials R(n,x) satisfy the recursion R(n+1,x) = (x+x^2)*R'(n,x) + x*R(n,x) where ' indicates differentiation with respect to x. - Philippe Deléham, Feb 11 2013

T(n,k) = [t^k] (n! [x^n] (1/(1-t*(exp(x)-1)))). - Peter Luschny, Jan 23 2017

EXAMPLE

The triangle T(n,k) begins:

n\k 0 1    2     3      4       5        6        7        8        9      10 ...

0:  1

1:  0 1

2:  0 1    2

3:  0 1    6     6

4:  0 1   14    36     24

5:  0 1   30   150    240     120

6:  0 1   62   540   1560    1800      720

7:  0 1  126  1806   8400   16800    15120     5040

8:  0 1  254  5796  40824  126000   191520   141120    40320

9:  0 1  510 18150 186480  834120  1905120  2328480  1451520   362880

10: 0 1 1022 55980 818520 5103000 16435440 29635200 30240000 16329600 3628800

... reformatted and extended. - Wolfdieter Lang, Mar 31 2017

MAPLE

A131689 := proc(n, k) combinat[stirling2](n, k)*k! end: # Peter Luschny, Sep 17 2011

# Alternatively:

A131689_row := proc(n) 1/(1-t*(exp(x)-1)); expand(series(%, x, n+1)); n!*coeff(%, x, n);

PolynomialTools:-CoefficientList(%, t) end:

for n from 0 to 9 do A131689_row(n) od; # Peter Luschny, Jan 23 2017

MATHEMATICA

t[n_, k_] := k!*StirlingS2[n, k]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 25 2014 *)

CROSSREFS

Case m=1 of the polynomials defined in A278073.

Cf. A000142 (diagonal), A000670 (row sums), A000012 (alternating row sums), A210029 (central terms).

Cf. A001286, A037960, A037961, A037962, A037963, A028246.

Cf. A008292, A028246 (o.g.f. and e.g.f. of sums of powers).

Sequence in context: A089949 A085845 A138106 * A278075 A114329 A241011

Adjacent sequences:  A131686 A131687 A131688 * A131690 A131691 A131692

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Sep 14 2007

EXTENSIONS

Formula corrected by Philippe Deléham, Feb 11 2013

STATUS

approved

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Last modified October 22 11:50 EDT 2017. Contains 293764 sequences.