Search: id:A008279
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%I A008279
%S A008279 1,1,1,1,2,2,1,3,6,6,1,4,12,24,24,1,5,20,60,120,120,1,6,30,
%T A008279 120,360,720,720,1,7,42,210,840,2520,5040,5040,1,8,56,336,
%U A008279 1680,6720,20160,40320,40320,1,9,72,504,3024,15120,60480
%N A008279 Triangle T(n,k) = n!/(n-k)! (0 <= k <= n) read by rows, giving number 
               of permutations of n things k at a time.
%C A008279 Also called permutation coefficients.
%C A008279 Also falling factorials triangle A068424 with column a(n,0)=1 and row 
               a(0,1)=1 else a(0,k)=0, added. - Wolfdieter Lang, Nov 07 2003
%C A008279 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 
               2009: (Start)
%C A008279 The higher order exponential integrals E(x,m,n) are defined in A163931 
               and for information about the asymptotic expansion of E(x,m=1,n) 
               see A130534. The asymptotic expansions for n = 1, 2, 3, 4, ... , 
               lead to the right hand columns of the triangle given above.
%C A008279 (End)
%D A008279 CRC Standard Mathematical Tables and Formulae, 30th ed., 1996, p. 176; 
               31st ed., p. 215, Section 3.3.11.1.
%D A008279 Maple V Reference Manual, p. 490, numbperm(n,k).
%H A008279 T. D. Noe, <a href="b008279.txt">Rows n=0..100 of triangle, flattened</
               a>
%H A008279 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               On generalizations of Stirling number triangles</a>, J. Integer Seqs., 
               Vol. 3 (2000), #00.2.4.
%F A008279 E.g.f.: sum T(n,k) x^n/n! y^k = exp(x)/(1-x*y). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Aug 19 2002
%F A008279 Equals A007318 * A136572 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 
               07 2008
%F A008279 T(n, k) = n*T(n-1, k-1) = k*T(n-1, k-1)+T(n-1, k) = n*T(n-1, k)/(n-k) 
               = (n-k+1)*T(n, k-1) - Henry Bottomley (se16(AT)btinternet.com), Mar 
               29 2001
%F A008279 T(n, k) = n!/(n-k)! if n >= k >= 0 else 0. G.f. for k-th column k!*x^k/
               (1-x)^(k+1), k >= 0. E.g.f. for n-th row (1+x)^n, n >= 0.
%F A008279 Sum T(n, k)x^k = permanent of n X n matrix a_ij = (x+1 if i=j, x otherwise). 
               - Michael Somos Mar 05 2004
%F A008279 Ramanujan psi_1(k, x) polynomials evaluated at n+1. - Ralf Stephan, Apr 
               16 2004
%F A008279 E.g.f. sum T(n,k) x^n/n! y^k/k! = e^{x+xy}. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), 
               Feb 07 2006
%F A008279 The triangle is the binomial transform of an infinite matrix with (1, 
               1, 2, 6, 24...) in the main diagonal and the rest zeros. - Gary W. 
               Adamson (qntmpkt(AT)yahoo.com), Nov 20 2006
%F A008279 G.f.: 1/(1-x-xy/(1-xy/(1-x-2xy/(1-2xy/(1-x-3xy/(1-3xy/(1-x-4xy/(1-4xy/
               (1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), 
               Feb 11 2009]
%e A008279 Triangle begins:
%e A008279 1
%e A008279 1, 1
%e A008279 1, 2, 2
%e A008279 1, 3, 6, 6
%e A008279 1, 4, 12, 24, 24
%e A008279 1, 5, 20, 60, 120, 120
%e A008279 1, 6, 30, 120, 360, 720, 720
%e A008279 1, 7, 42, 210, 840, 2520, 5040, 5040
%e A008279 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320
%e A008279 1, 9, 72, 504, 3024, 15120, 60480, 181440, 362880, 362880
%e A008279 1, 10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800, 3628800
%p A008279 with(combstruct):for n from 0 to 10 do seq(count(Permutation(n),size=m), 
               m = 0 .. n) od; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Dec 16 2007
%o A008279 (PARI) T(n,k)=if(k<0|k>n,0,n!/(n-k)!)
%o A008279 (PARI) T(n,k)=local(A,p);if(k<0|k>n,0,if(n==0,1, A=matrix(n,n,i,j,x+(i==j));
               polcoeff(sum(i=1,n!,if(p=numtoperm(n,i),prod(j=1,n,A[j,p[j]]))),k)))
%Y A008279 Row sums give A000522.
%Y A008279 Cf. A001497, A001498, A136572.
%Y A008279 Sequence in context: A082037 A163649 A110858 this_sequence A056043 A158497 
               A110564
%Y A008279 Adjacent sequences: A008276 A008277 A008278 this_sequence A008280 A008281 
               A008282
%K A008279 nonn,tabl,nice,easy
%O A008279 0,5
%A A008279 N. J. A. Sloane (njas(AT)research.att.com).

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