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A054654
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Triangle of Stirling numbers of 1st kind, S(n, n-k), n >= 0, 0 <= k <= n.
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12
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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, 175, -735, 1624, -1764, 720, 0, 1, -28, 322, -1960, 6769, -13132, 13068, -5040, 0
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OFFSET
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0,8
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COMMENTS
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Triangle is the matrix product of the binomial coefficients with the Stirling numbers of the first kind.
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.
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).
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
From Doudou Kisabaka (dougk7(AT)gmail.com), Dec 18 2009: (Start)
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.
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. (End)
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LINKS
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Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Eric Weisstein's World of Mathematics, Pochhammer Symbol
Eric Weisstein's World of Mathematics, Rising Factorial
Eric Weisstein's World of Mathematics, FallingFactorial
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FORMULA
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n!*binomial(x, n) = Sum_{k=0..n} T(n, k)*x^(n-k).
(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.
T(n, k) = (-1)^k*Sum_{j=0..k} E2(k, j)*binomial(n+j-1, 2*k), where E2(k, j) are the second-order Eulerian numbers A340556. - Peter Luschny, Feb 21 2021
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EXAMPLE
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Matrix begins:
1, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 1, -1, 2, -6, 24, -120, 720, -5040, ...
0, 0, 1, -3, 11, -50, 274, -1764, 13068, ...
0, 0, 0, 1, -6, 35, -225, 1624, -13132, ...
0, 0, 0, 0, 1, -10, 85, -735, 6769, ...
0, 0, 0, 0, 0, 1, -15, 175, -1960, ...
0, 0, 0, 0, 0, 0, 1, -21, 322, ...
0, 0, 0, 0, 0, 0, 0, 1, -28, ...
0, 0, 0, 0, 0, 0, 0, 0, 1, ...
...
Triangle begins:
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, 175, -735, 1624, -1764, 720, 0;
...
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MAPLE
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a054654_row := proc(n) local k; seq(coeff(expand((-1)^n*pochhammer (-x, n)), x, n-k), k=0..n) end: # Peter Luschny, Nov 28 2010
seq(seq(Stirling1(n, n-k), k=0..n), n=0..8); # Peter Luschny, Feb 21 2021
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MATHEMATICA
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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 *)
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PROG
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(PARI) T(n, k)=polcoeff(n!*binomial(x, n), n-k)
(Haskell)
a054654 n k = a054654_tabl !! n !! k
a054654_row n = a054654_tabl !! n
a054654_tabl = map reverse a048994_tabl
-- Reinhard Zumkeller, Mar 18 2014
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CROSSREFS
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Essentially Stirling numbers of first kind, multiplied by factorials - see A008276.
The Stirling2 counterpart is A106800.
Cf. A054655, A039810, A039814, A126350, A126351, A126353, A340556.
Sequence in context: A081576 A330785 A292717 * A253669 A154477 A322324
Adjacent sequences: A054651 A054652 A054653 * A054655 A054656 A054657
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KEYWORD
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tabl,sign,easy,nice,changed
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AUTHOR
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N. J. A. Sloane, Apr 18 2000
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EXTENSIONS
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Additional comments from Thomas Wieder, Dec 29 2006
Edited by N. J. A. Sloane at the suggestion of Eric W. Weisstein, Jan 20 2008
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STATUS
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approved
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