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 A008306 Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)). 15
 1, 2, 6, 3, 24, 20, 120, 130, 15, 720, 924, 210, 5040, 7308, 2380, 105, 40320, 64224, 26432, 2520, 362880, 623376, 303660, 44100, 945, 3628800, 6636960, 3678840, 705320, 34650, 39916800, 76998240, 47324376, 11098780, 866250, 10395 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 COMMENTS Also, T(n,k) = number of derangements of {1..n} with k cycles. Also, T(n,k) = number of permutations of {1..n} with k cycles of length >= 2. The sum of the n-th row is the n-th subfactorial: A000166(n). - Gary Detlefs, Jul 14 2010 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 = 2..125 of table, flattened J. Fernando Barbero G., Jesús Salas, Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013. W. Carlitz, On some polynomials of Tricomi, Bollettino dell'Unione Matematica Italiana, Serie 3, Vol. 13, (1958), n. 1, p. 58-64 W. Gautschi, The incomplete gamma functions since Tricomi (Cf. p. 206-207.) P. Gniewek, B. Jeziorski, Convergence properties of the multipole expansion of the exchange contribution to the interaction energy, arXiv preprint arXiv:1601.03923 [physics.chem-ph], 2016. S. Karlin and J. McGregor, Many server queuing processes with Poisson input and exponential service times, Pacific Journal of Mathematics, Vol. 8, No. 1, p. 87-118, March (1958)   (Cf. p. 117) R. Paris, A uniform asymptotic expansion for the incomplete gamma function, Journal of Computational and Applied Mathematics, 148 (2002), p. 223-239 (See 333. From Tom Copeland, Jan 03 2016) M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq. 14 (2011) # 11.9.7. A. Topuzoglu, The Carlitz rank of permutations of finite fields: A survey, Journal of Symbolic Computation, Online, Dec 07, 2013. Eric Weisstein's World of Mathematics, Permutation Cycle Eric Weisstein's World of Mathematics, Stirling Number of the First Kind FORMULA T(n,k) = Sum_{i=0..k} (-1)^i * binomial(n,i) * |stirling1(n-i,k-i)| = (-1)^(n+k) * Sum_{i=0..k} (-1)^i * binomial(n,i) * A008275(n-i,k-i). - Max Alekseyev, Sep 08 2018 E.g.f.: 1 + Sum_{1 <= 2k <= n} T(n, k)*t^n*u^k/n! = exp(-t*u)*(1-t)^(-u). Recurrence: T(n, k) = (n-1)*(T(n-1, k) + T(n-2, k-1)) for 1 <= k <= n/2 with boundary conditions T(0,0) = 1, T(n,0) = 0 for n >= 1, T(n,k) = 0 for k > n/2. - David Callan, May 16 2005 E.g.f. for column k: B(A(x)) where A(x) = log(1/1-x) - x and B(x) = x^k/k! From Tom Copeland, Jan 05 2016: (Start) This signed array's row polynomials are the orthogonal NL(n,x;x-n) = n! Sum_{k=0..n} binomial(x,n-k) (-x)^k/k!, the normalized Laguerre polynomials of order (x-n) as discussed in Gautschi (the Temme, Carlitz, and Karlin and McGregor references come from this paper) in regard to asymptotic expansions of the upper incomplete gamma function--Tricomi's Cinderella of special functions. e^(xt) (1-t)^x = Sum_{n>=0} NL(n,x;x-n) x^n/n!. The first few are NL(0,x) = 1 NL(1,x) = 0 NL(2,x) = -x NL(3,x) = 2x NL(4,x) = -6x + 3x^2. With D=d/dx, :xD:^n = x^n D^n, :Dx:^n = D^n x^n, and K(a,b,c), the Kummer confluent hypergeometric function, NL(n,x;y-n) = n! e^x binomial(xD+y,n) e^(-x) = n! e^x Sum_{k=0..n} binomial(k+y,n) (-x)^k/k! = e^x x^(-y+n) D^n (x^y e^(-x)) = e^x x^(-y+n) :Dx:^n x^(y-n) e^(-x) = e^x x^(-y+n) n! L(n,:xD:,0) x^(y-n) e^(-x) = n! binomial(y,n) K(-n,y-n+1,x) = n! e^x (-1)^n binomial(-xD-y+n-1,n) e^(-x). Evaluate these expressions at y=x after the derivative operations to obtain NL(n,x;x-n). (End) EXAMPLE Rows 2 through 7 are:     1;     2;     6,   3;    24,  20;   120, 130,  15;   720, 924, 210; MAPLE A008306 := proc(n, k) local j; add(binomial(j, n-2*k)*A008517(n-k, j), j=0..n-k) end; seq(print(seq(A008306(n, k), k=1..iquo(n, 2))), n=2..12): # Peter Luschny, Apr 20 2011 MATHEMATICA t[0, 0] = 1; t[n_, 0] = 0; t[n_, k_] /; k > n/2 = 0; t[n_, k_] := t[n, k] = (n - 1)*(t[n - 1, k] + t[n - 2, k - 1]); A008306 = Flatten[ Table[ t[n, k], {n, 2, 12}, {k, 1, Quotient[n, 2]}]] (* Jean-François Alcover, Jan 25 2012, after David Callan *) PROG (PARI) { A008306(n, k) = (-1)^(n+k) * sum(i=0, k, (-1)^i * binomial(n, i) * stirling(n-i, k-i, 1) ); } \\ Max Alekseyev, Sep 08 2018 (Haskell) a008306 n k = a008306_tabf !! (n-2) !! (k-1) a008306_row n = a008306_tabf !! (n-2) a008306_tabf = map (fst . fst) \$ iterate f ((, ), 3) where    f ((us, vs), x) =      ((vs, map (* x) \$ zipWith (+) ( ++ us) (vs ++ )), x + 1) -- Reinhard Zumkeller, Aug 05 2013 CROSSREFS Cf. A000166, A106828 (another version), A079510 (rearranged triangle), A235706 (specializations). Diagonals: A000142, A000276, A000483. Diagonals give reversed rows of A111999. Sequence in context: A206493 A304085 A302783 * A231171 A331431 A248120 Adjacent sequences:  A008303 A008304 A008305 * A008307 A008308 A008309 KEYWORD tabf,nonn,nice,easy AUTHOR EXTENSIONS More terms from Larry Reeves (larryr(AT)acm.org), Feb 16 2001 STATUS approved

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Last modified March 29 17:23 EDT 2020. Contains 333116 sequences. (Running on oeis4.)