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 A106828 Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 0 and 0 <= k <= floor(n/2)). 2
 1, 0, 0, 1, 0, 2, 0, 6, 3, 0, 24, 20, 0, 120, 130, 15, 0, 720, 924, 210, 0, 5040, 7308, 2380, 105, 0, 40320, 64224, 26432, 2520, 0, 362880, 623376, 303660, 44100, 945, 0, 3628800, 6636960, 3678840, 705320, 34650, 0, 39916800, 76998240, 47324376, 11098780, 866250, 10395 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Another version of the triangle is in A008306. A signed version of this triangle is given by the exponential Riordan array [1, log(1+t)-t]. Its row sums are (-1)^n*(1-n). Another version is [1, log(1-t)+t], whose row sums are 1-n. - Paul Barry, May 10 2008 REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256. J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 75. LINKS Reinhard Zumkeller, Rows n = 0..124 of table, flattened Feng-Zhen Zhao, Some Properties of Associated Stirling Numbers, Journal of Integer Sequences, Article 08.1.7, 2008. FORMULA T(n, k) = Sum_{j=0..n-k} binomial(j, n-2*k)*E2(n-k, j), where E2 are the second-order Eulerian numbers (A008517). - Peter Luschny, Jan 13 2016 Also the Bell transform of the sequence g(k) = k! if k>0 else 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 13 2016 EXAMPLE Rows 0 though 7 are:   1;   0,   0,   1;   0,   2,   0,   6,   3;   0,  24,  20,   0, 120, 130,  15;   0, 720, 924, 210; MAPLE A106828 := (n, k) -> add(binomial(j, n-2*k)*combinat:-eulerian2(n-k, j), j=0..n-k): seq(print(seq(A106828(n, k), k=0..iquo(n, 2))), n=0..9); # Peter Luschny, Apr 20 2011 (revised Jan 13 2016) MATHEMATICA Eulerian2[n_, k_] := Eulerian2[n, k] = If[k == 0, 1, If[k == n, 0, Eulerian2[n - 1, k] (k + 1) + Eulerian2[n - 1, k - 1] (2 n - k - 1)]]; T[n_, k_] := Sum[Binomial[j, n - 2 k] Eulerian2[n - k, j], {j, 0, n - k}]; Table[T[n, k], {n, 0, 12}, {k, 0, n/2}] (* Jean-François Alcover, Jun 13 2019 *) PROG (Haskell) a106828 n k = a106828_tabf !! n !! k a106828_row n = a106828_tabf !! n a106828_tabf = map (fst . fst) \$ iterate f (([1], [0]), 1) where    f ((us, vs), x) =      ((vs, map (* x) \$ zipWith (+) ([0] ++ us) (vs ++ [0])), x + 1) -- Reinhard Zumkeller, Aug 05 2013 (Sage) # The function bell_transform is defined in A264428. # Computes the full triangle 0<=k<=n. def A106828_row(n):     g = lambda k: factorial(k) if k>0 else 0     s = [g(k) for k in (0..n)]     return bell_transform(n, s) [A106828_row(n) for n in (0..8)] # Peter Luschny, Jan 13 2016 CROSSREFS See A008306 for more information. Cf. A008619 (row lengths), A000166 (row sums). Sequence in context: A054877 A269795 A095834 * A055302 A055349 A161174 Adjacent sequences:  A106825 A106826 A106827 * A106829 A106830 A106831 KEYWORD tabf,nonn,easy AUTHOR N. J. A. Sloane, May 22 2005 EXTENSIONS Removed extra 0 in row 1 from Michael Somos, Jan 19 2011 STATUS approved

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Last modified October 22 03:04 EDT 2019. Contains 328315 sequences. (Running on oeis4.)