%I A054654
%S A054654 1,1,0,1,1,0,1,3,2,0,1,6,11,6,0,1,10,35,50,24,0,1,15,85,225,274,120,0,
               1,
%T A054654 21,175,735,1624,1764,720,0,1,28,322,1960,6769,13132,13068,5040,0
%V A054654 1,1,0,1,-1,0,1,-3,2,0,1,-6,11,-6,0,1,-10,35,-50,24,0,1,-15,85,-225,274,
               -120,0,1,-21,
%W A054654 175,-735,1624,-1764,720,0,1,-28,322,-1960,6769,-13132,13068,-5040,0
%N A054654 Triangle read by rows: matrix product of the binomial coefficients with 
               the Stirling numbers of the first kind.
%C A054654 The sum of the entries on each row of the triangle, starting on the 3rd 
               row, equals 0. E.g. 1+(-3)+2+0 = 0
%C A054654 The entries on the triangle can be computed as follows. T(n,r) = T(n-1,
               r) - (n-1)*T(n-1,r-1). T(n,r) = 0 when r equals 0 or r > n. T(n,r) 
               = 1 if n==1
%C A054654 Triangle T(n,k) giving coefficients in expansion of n!*C(x,n) in powers 
               of x. E.g. 3!*C(x,3) = x^3-3*x^2+2*x.
%C A054654 The matrix product of binomial coefficients with the Stirling numbers 
               of the first kind results in the Stirling numbers of the first kind 
               again, but the triangle is shifted by (1,1).
%C A054654 Essentially [1,0,1,0,1,0,1,0,...] DELTA [0,-1,-1,-2,-2,-3,-3,-4,-4,...] 
               where DELTA is the operator defined in A084938 ; mirror image of 
               the Stirling-1 triangle A048994 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Dec 30 2006
%H A054654 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PochhammerSymbol.html">Pochhammer Symbol</a>
%H A054654 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RisingFactorial.html">Rising Factorial</a>
%H A054654 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               FallingFactorial.html">FallingFactorial</a>
%F A054654 n!*binomial(x, n) = Sum T(n, k)*x^(n-k), k=0..n.
%F A054654 (In Maple notation:) Matrix product A*B of matrix A[i,j]:=binomial(j-1,
               i-1) with i = 1 to p+1, j = 1 to p+1, p=8 and of matrix B[i,j]:=stirling1(j,
               i) with i from 1 to d, j from 1 to d, d=9.
%e A054654 Matrix begins:
%e A054654 1 0 0 0 0 0 0 0 0
%e A054654 0 1 -1 2 -6 24 -120 720 -5040
%e A054654 0 0 1 -3 11 -50 274 -1764 13068
%e A054654 0 0 0 1 -6 35 -225 1624 -13132
%e A054654 0 0 0 0 1 -10 85 -735 6769
%e A054654 0 0 0 0 0 1 -15 175 -1960
%e A054654 0 0 0 0 0 0 1 -21 322
%e A054654 0 0 0 0 0 0 0 1 -28
%e A054654 0 0 0 0 0 0 0 0 1
%e A054654 ...
%e A054654 Triangle begins:
%e A054654 1;
%e A054654 1, 0;
%e A054654 1, -1, 0;
%e A054654 1, -3, 2, 0;
%e A054654 1, -6, 11, -6, 0;
%e A054654 1, -10, 35, -50, 24, 0;
%e A054654 1, -15, 85, -225, 274, -120, 0;
%e A054654 1, -21, 175, -735, 1624, -1764, 720, 0;
%e A054654 ...
%o A054654 (PARI) T(n,k)=polcoeff(n!*binomial(x,n), n-k)
%Y A054654 Essentially Stirling numbers of first kind, multiplied by factorials 
               - see A008276. Cf. A054655.
%Y A054654 Cf. A039810, A039814, A126350, A126351, A126353.
%Y A054654 Sequence in context: A008783 A139144 A081576 this_sequence A154477 A142071 
               A118972
%Y A054654 Adjacent sequences: A054651 A054652 A054653 this_sequence A054655 A054656 
               A054657
%K A054654 tabl,sign,easy,nice,new
%O A054654 0,8
%A A054654 N. J. A. Sloane (njas(AT)research.att.com), Apr 18 2000
%E A054654 Additional comments from Thomas Wieder (thomas.wieder(AT)t-online.de), 
               Dec 29 2006
%E A054654 Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion 
               of Eric Weisstein, Jan 20 2008
%E A054654 Added a comment concerning the sum of the entries on a row which is always 
               0 for all row >= 3 and the formula T(n,r)=T(n-1,r) - (n-1)*T(n-1,
               r-1) Doudou Kisabaka (dougk7(AT)gmail.com), Dec 18 2009