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%I #95 Dec 23 2020 07:41:19
%S 1,2,6,3,24,20,120,130,15,720,924,210,5040,7308,2380,105,40320,64224,
%T 26432,2520,362880,623376,303660,44100,945,3628800,6636960,3678840,
%U 705320,34650,39916800,76998240,47324376,11098780,866250,10395
%N Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).
%C Also, T(n,k) is the number of derangements (permutations with no fixed points) of {1..n} with k cycles.
%C The sum of the n-th row is the n-th subfactorial: A000166(n). - _Gary Detlefs_, Jul 14 2010
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 75.
%H Reinhard Zumkeller, <a href="/A008306/b008306.txt">Rows n = 2..125 of table, flattened</a>
%H J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, <a href="http://arxiv.org/abs/1307.2010">Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure</a>, arXiv:1307.2010 [math.CO], 2013.
%H W. Carlitz, <a href="http://www.bdim.eu/item?fmt=pdf&id=BUMI_1958_3_13_1_58_0">On some polynomials of Tricomi</a>, Bollettino dell'Unione Matematica Italiana, Serie 3, Vol. 13, (1958), n. 1, p. 58-64
%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>
%H W. Gautschi, <a href="http://www.cs.purdue.edu/homes/wxg/selected_works/section_02/155.pdf">The incomplete gamma functions since Tricomi</a> (Cf. p. 206-207.)
%H P. Gniewek and B. Jeziorski, <a href="https://arxiv.org/abs/1601.03923">Convergence properties of the multipole expansion of the exchange contribution to the interaction energy</a>, arXiv preprint arXiv:1601.03923 [physics.chem-ph], 2016.
%H S. Karlin and J. McGregor, <a href="http://msp.org/pjm/1958/8-1/pjm-v8-n1-p08-p.pdf">Many server queuing processes with Poisson input and exponential service times</a>, Pacific Journal of Mathematics, Vol. 8, No. 1, p. 87-118, March (1958). Cf. p. 117.
%H R. Paris, <a href="http://dx.doi.org/10.1016/S0377-0427(02)00553-8">A uniform asymptotic expansion for the incomplete gamma function</a>, Journal of Computational and Applied Mathematics, 148 (2002), p. 223-239. (See 333. From Tom Copeland, Jan 03 2016)
%H M. Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Spivey/spivey31.html">On Solutions to a General Combinatorial Recurrence</a>, J. Int. Seq. 14 (2011) # 11.9.7.
%H N. Temme, <a href="https://ir.cwi.nl/pub/6455">A class of polynomials related to those of Laguerre</a>
%H N. Temme, <a href="https://ir.cwi.nl/pub/2488">Traces to Tricomi in recent work on special functions and asymptotics of integrals</a>
%H A. Topuzoglu, <a href="https://doi.org/10.1016/j.jsc.2013.07.004">The Carlitz rank of permutations of finite fields: A survey</a>, Journal of Symbolic Computation, Online, Dec 07, 2013.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PermutationCycle.html">Permutation Cycle</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html">Stirling Number of the First Kind</a>
%H Shawn L. Witte, <a href="https://www.math.ucdavis.edu/~tdenena/dissertations/201910_Witte_Dissertation.pdf">Link Nomenclature, Random Grid Diagrams, and Markov Chain Methods in Knot Theory</a>, Ph. D. Dissertation, University of California-Davis (2020).
%F 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
%F E.g.f.: 1 + Sum_{1 <= 2*k <= n} T(n, k)*t^n*u^k/n! = exp(-t*u)*(1-t)^(-u).
%F 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, and T(n,k) = 0 for k > n/2. - _David Callan_, May 16 2005
%F E.g.f. for column k: B(A(x)) where A(x) = log(1/1-x)-x and B(x) = x^k/k!.
%F From _Tom Copeland_, Jan 05 2016: (Start)
%F The row polynomials of this signed array 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.
%F e^(x*t)*(1-t)^x = Sum_{n>=0} NL(n,x;x-n)*x^n/n!.
%F The first few are
%F NL(0,x) = 1
%F NL(1,x) = 0
%F NL(2,x) = -x
%F NL(3,x) = 2*x
%F NL(4,x) = -6*x + 3*x^2.
%F 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)
%e Rows 2 through 7 are:
%e 1;
%e 2;
%e 6, 3;
%e 24, 20;
%e 120, 130, 15;
%e 720, 924, 210;
%p A008306 := proc(n,k) local j;
%p add(binomial(j,n-2*k)*A008517(n-k,j),j=0..n-k) end;
%p seq(print(seq(A008306(n,k),k=1..iquo(n,2))),n=2..12):
%p # _Peter Luschny_, Apr 20 2011
%t 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_ *)
%o (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
%o (Haskell)
%o a008306 n k = a008306_tabf !! (n-2) !! (k-1)
%o a008306_row n = a008306_tabf !! (n-2)
%o a008306_tabf = map (fst . fst) $ iterate f (([1], [2]), 3) where
%o f ((us, vs), x) =
%o ((vs, map (* x) $ zipWith (+) ([0] ++ us) (vs ++ [0])), x + 1)
%o -- _Reinhard Zumkeller_, Aug 05 2013
%Y Cf. A000166, A106828 (another version), A079510 (rearranged triangle), A235706 (specializations).
%Y Diagonals: A000142, A000276, A000483.
%Y Diagonals give reversed rows of A111999.
%K tabf,nonn,nice,easy
%O 2,2
%A _N. J. A. Sloane_
%E More terms from Larry Reeves (larryr(AT)acm.org), Feb 16 2001
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