0,5

Also called permutation coefficients.

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

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

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.

(End)

CRC Standard Mathematical Tables and Formulae, 30th ed., 1996, p. 176; 31st ed., p. 215, Section 3.3.11.1.

Maple V Reference Manual, p. 490, numbperm(n,k).

T. D. Noe, Rows n=0..100 of triangle, flattened

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

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

Equals A007318 * A136572 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 07 2008

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

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.

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

Ramanujan psi_1(k, x) polynomials evaluated at n+1. - Ralf Stephan, Apr 16 2004

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

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

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]

Triangle begins:

1

1, 1

1, 2, 2

1, 3, 6, 6

1, 4, 12, 24, 24

1, 5, 20, 60, 120, 120

1, 6, 30, 120, 360, 720, 720

1, 7, 42, 210, 840, 2520, 5040, 5040

1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320

1, 9, 72, 504, 3024, 15120, 60480, 181440, 362880, 362880

1, 10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800, 3628800

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

(PARI) T(n, k)=if(k<0|k>n, 0, n!/(n-k)!)

(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)))

Row sums give A000522.

Cf. A001497, A001498, A136572.

Sequence in context: A082037 A163649 A110858 this_sequence A056043 A158497 A110564

Adjacent sequences: A008276 A008277 A008278 this_sequence A008280 A008281 A008282

nonn,tabl,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).