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A054654 Triangle read by rows: matrix product of the binomial coefficients with the Stirling numbers of the first kind. 11

%I

%S 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,

%T 274,-120,0,1,-21,175,-735,1624,-1764,720,0,1,-28,322,-1960,6769,

%U -13132,13068,-5040,0

%N Triangle read by rows: matrix product of the binomial coefficients with the Stirling numbers of the first kind.

%C 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 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 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 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 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 Deléham_, Dec 30 2006

%H Reinhard Zumkeller, <a href="/A054654/b054654.txt">Rows n = 0..125 of triangle, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RisingFactorial.html">Rising Factorial</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FallingFactorial.html">FallingFactorial</a>

%F n!*binomial(x, n) = Sum T(n, k)*x^(n-k), k=0..n.

%F (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 Matrix begins:

%e 1 0 0 0 0 0 0 0 0

%e 0 1 -1 2 -6 24 -120 720 -5040

%e 0 0 1 -3 11 -50 274 -1764 13068

%e 0 0 0 1 -6 35 -225 1624 -13132

%e 0 0 0 0 1 -10 85 -735 6769

%e 0 0 0 0 0 1 -15 175 -1960

%e 0 0 0 0 0 0 1 -21 322

%e 0 0 0 0 0 0 0 1 -28

%e 0 0 0 0 0 0 0 0 1

%e ...

%e Triangle begins:

%e 1;

%e 1, 0;

%e 1, -1, 0;

%e 1, -3, 2, 0;

%e 1, -6, 11, -6, 0;

%e 1, -10, 35, -50, 24, 0;

%e 1, -15, 85, -225, 274, -120, 0;

%e 1, -21, 175, -735, 1624, -1764, 720, 0;

%e ...

%p a054654_row := proc(n) local k; seq(coeff(expand((-1)^n*pochhammer (-x,n)),x,n-k),k=0..n) end:

%t row[n_] := Reverse[ CoefficientList[ (-1)^n*Pochhammer[-x, n], x] ]; Flatten[ Table[ row[n], {n, 0, 8}]] (* _Jean-François Alcover_, Feb 16 2012, after Maple *)

%o (PARI) T(n,k)=polcoeff(n!*binomial(x,n), n-k)

%o (Haskell)

%o a054654 n k = a054654_tabl !! n !! k

%o a054654_row n = a054654_tabl !! n

%o a054654_tabl = map reverse a048994_tabl

%o -- _Reinhard Zumkeller_, Mar 18 2014

%Y Essentially Stirling numbers of first kind, multiplied by factorials - see A008276. Cf. A054655.

%Y Cf. A039810, A039814, A126350, A126351, A126353.

%K tabl,sign,easy,nice

%O 0,8

%A _N. J. A. Sloane_, Apr 18 2000

%E Additional comments from _Thomas Wieder_, Dec 29 2006

%E Edited by _N. J. A. Sloane_ at the suggestion of Eric Weisstein, Jan 20 2008

%E 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

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Last modified February 21 04:18 EST 2019. Contains 320371 sequences. (Running on oeis4.)