%I A008276
%S A008276 1,1,1,1,3,2,1,6,11,6,1,10,35,50,24,1,15,85,225,274,120,1,
%T A008276 21,175,735,1624,1764,720,1,28,322,1960,6769,13132,13068,
%U A008276 5040,1,36,546,4536,22449,67284,118124,109584,40320,1,45
%V A008276 1,1,-1,1,-3,2,1,-6,11,-6,1,-10,35,-50,24,1,-15,85,-225,274,-120,1,
%W A008276 -21,175,-735,1624,-1764,720,1,-28,322,-1960,6769,-13132,13068,
%X A008276 -5040,1,-36,546,-4536,22449,-67284,118124,-109584,40320,1,-45
%N A008276 Triangle of Stirling numbers of first kind, s(n,n-k+1), n >= 1, 1<=k<=n. 
               Also triangle T(n,k) giving coefficients in expansion of n!*C(x,n)/
               x in powers of x.
%C A008276 n-th row of the triangle = charpoly of an (n-1)x(n-1) matrix with (1,
               2,3,...) in the diagonal and the rest zeros. [From Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Mar 19 2009]
%D A008276 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 833.
%D A008276 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied 
               Tables, Cambridge, 1966, p. 226.
%D A008276 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd 
               ed. (Addison-Wesley, 1994), p. 257.
%H A008276 T. D. Noe, <a href="b008276.txt">Rows n=0..100 of triangle, flattened</
               a>
%H A008276 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A008276 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">
               Sobalian Coefficients</a>.
%H A008276 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/
               index.html">Miscellaneous</a>.
%F A008276 n!*binomial(x, n)= Sum T(n, k)*x^(n-k), k=1..n-1.
%F A008276 |A008276(n, k)| = T(n-1, k-1) where T(n, k) is the triangle, read by 
               rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 1, 2, 2, 
               3, 3, 4, 4, 5, 5, ...]; A008276(n, k) = T(n-1, k-1) where T(n, k) 
               is the triangle, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 
               1, ...] DELTA [ -1, -1, -2, -2, -3, -3, -4, -4, -5, -5, ...]. Here 
               DELTA is the operator defined in A084938 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Dec 30 2003
%F A008276 |T(n, k)| = sum(A008517(k, m+1)*binomial(n+m, 2*(k-1)), m=0..n), n>=k>
               =1. A008517 is the second-order Eulerian triangle. See the Graham 
               et al. reference p. 257, eq. (6.44).
%F A008276 A111999 formula for signed T(n, k).
%F A008276 |T(n, k)| = sum(A112486(k-1, m)*binomial(n-1, k-1+m), m=0..min(k-1, n-k)) 
               if n>=k>=1, else 0. - W. Lang Sep 12 2005, see A112486.
%F A008276 |T(n, k)| = (f(n-1, k-1)/(2*(k-1))!)* sum(A112486(k-1, m)*f(2*(k-1), 
               k-1-m)*f(n-k, m), m=0..min(k-1, n-k)) if n>=k>=1, else 0, where f(n, 
               k) stands for the falling factorial n*(n-1)*...*(n-(k-1)) and f(n, 
               0):=1. - W. Lang Sep 12 2005, see A112486.
%F A008276 With P(n,t) = sum(k=0,...,n-1) T(n,k+1) * t^k = (1-t) (1-2t)...(1-(n-1)t) 
               and P(0,t) = 1, exp[P(.,t)*x] = (1+tx)^(1/t) . Compare A094638. T(n,
               k+1) = (1/k!) (D_t)^k (D_x)^n [ (1+tx)^(1/t) - 1 ] evaluated at t=x=0 
               . - Tom Copeland (tcjpn(AT)msn.com), Dec 09 2007
%F A008276 PRODUCT((x-i): 1<=i<=n) = SUM(T(n,k)*x^k: 0<=k<=n). - Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Dec 29 2007
%e A008276 3!*C(x,3) = x^3-3*x^2+2*x.
%e A008276 1; 1,-1; 1,-3,2; 1,-6,11,-6; 1,-10,35,-50,24; ...
%o A008276 (PARI) T(n,k)=if(n<1,0,n!*polcoeff(binomial(x,n),n-k+1))
%o A008276 (PARI) T(n,k)=if(n<1,0,n!*polcoeff(polcoeff(y*(1+y*x+x*O(x^n))^(1/y),
               n),k))
%Y A008276 See A008275 and A048994, which are the main entries for this triangle 
               of numbers. Cf. A054654, A054655.
%Y A008276 Cf. A084938, A145324.
%Y A008276 Cf. A094216, A008275, A003422, A000166, A000110, A000204, A000045, A000108.
%Y A008276 Sequence in context: A088617 A144250 A156367 this_sequence A094638 A143778 
               A164645
%Y A008276 Adjacent sequences: A008273 A008274 A008275 this_sequence A008277 A008278 
               A008279
%K A008276 sign,tabl,nice
%O A008276 1,5
%A A008276 N. J. A. Sloane (njas(AT)research.att.com).