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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). 41
 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 T(n,k) = the number of alignments of length k of n strings each of length 1. See Slowinski. An example is given below. Cf. A122193 (alignments of strings of length 2) and A299041 (alignments of strings of length 3). - Peter Bala, Feb 04 2018 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. M. Dukes, C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016. 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. J. B. Slowinski, The Number of Multiple Alignments, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:10.1006/mpev.1998.0522 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. [corrected by Philippe Deléham, Feb 11 2013] 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 The n-th row polynomial has the form x o x o ... o x (n factors), where o denotes the black diamond multiplication operator of Dukes and White. See also Bala, Example E8. - Peter Bala, Jan 08 2018 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 From Peter Bala, Feb 04 2018: (Start) T(4,2) = 14 alignments of length 2 of 4 strings of length 1. Examples include   (i) A -    (ii) A -    (iii) A -       B -         B -          - B       C -         - C          - C       - D         - D          - D There are C(4,1) = 4 alignments of type (i) with a single gap character - in column 1, C(4,2) = 6 alignments of type (ii) with two gap characters in column 1 and C(4,3) = 4 alignments of type (iii) with three gap characters in column 1, giving a total of 4 + 6 + 4 = 14 alignments. (End) MAPLE A131689 := (n, k) -> Stirling2(n, k)*k!: # 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 *) T[n_, k_] := If[n <= 0 || k <= 0, Boole[n == 0 && k == 0], Sum[(-1)^(i + k) Binomial[k, i] i^(n + k), {i, 0, k}]]; (* Michael Somos, Jul 08 2018 *) PROG (PARI) {T(n, k) = if( n<0, 0, sum(i=0, k, (-1)^(k + i) * binomial(k, i) * i^n))}; /* Michael Somos, Jul 08 2018 */ 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). Cf. A019538, A122193, A299041. Sequence in context: A089949 A085845 A138106 * A278075 A114329 A241011 Adjacent sequences:  A131686 A131687 A131688 * A131690 A131691 A131692 KEYWORD nonn,tabl,easy AUTHOR Philippe Deléham, Sep 14 2007 STATUS approved

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Last modified August 16 19:58 EDT 2018. Contains 313809 sequences. (Running on oeis4.)