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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). 27
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)

REFERENCES

Kreweras, Germain. "Une dualité élémentaire souvent utile dans les problèmes combinatoires." Mathématiques et Sciences Humaines 3 (1963): 31-41. Available from http://archive.numdam.org/ARCHIVE/MSH/MSH_1963__3_/MSH_1963__3__31_0/MSH_1963__3__31_0.pdf

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.

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

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<=k<=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<=k<=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_{0<=k<=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..inf} 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

Triangle begins:

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, 1451520, 362880 ;...

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.

Sequence in context: A089949 A085845 A138106 * A278075 A114329 A241011

Adjacent sequences:  A131686 A131687 A131688 * A131690 A131691 A131692

KEYWORD

nonn,tabl,changed

AUTHOR

Philippe Deléham, Sep 14 2007

EXTENSIONS

Formula corrected by Philippe Deléham, Feb 11 2013

STATUS

approved

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Last modified March 27 22:02 EDT 2017. Contains 284182 sequences.