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%I #28 Jun 28 2015 14:29:10
%S 1,-3,1,12,-7,1,-60,47,-12,1,360,-342,119,-18,1,-2520,2754,-1175,245,
%T -25,1,20160,-24552,12154,-3135,445,-33,1,-181440,241128,-133938,
%U 40369,-7140,742,-42,1,1814400,-2592720,1580508,-537628
%N Generalized Stirling number triangle of first kind.
%C a(n,m)= ^3P_n^m in the notation of the given reference with a(0,0) := 1. The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(3+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1.
%C In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer polynomials for (exp(3*t),exp(t)-1).
%C See A143492 for the unsigned version of this array and A143495 for the inverse. - _Peter Bala_, Aug 25 2008
%D Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.
%H Reinhard Zumkeller, <a href="/A049458/b049458.txt">Rows n = 0..125 of triangle, flattened</a>
%F a(n, m)= a(n-1, m-1) - (n+2)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n<m; a(n, -1) := 0, a(0, 0)=1. E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^3).
%F Triangle (signed) = [ -3, -1, -4, -2, -5, -3, -6, -4, -7, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; triangle (unsigned) = [3, 1, 4, 2, 5, 3, 6, 4, 7, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deléham's operator defined in A084938 (unsigned version in A143492).
%F E.g.f.: (1+y)^(x-3). - _Vladeta Jovovic_, May 17 2004
%F If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then T(n,i) = f(n,i,3), for n=1,2,...;i=0...n. - _Milan Janjic_, Dec 21 2008
%e 1;
%e -3, 1;
%e 12, -7, 1;
%e -60, 47, -12, 1;
%e 360, -342, 119, -18, 1;
%e s(2,x) = 12-7*x+x^2. S1(2,x) = -x+x^2 (Stirling1 polynomial).
%p A049458_row := n -> seq((-1)^(n-k)*coeff(expand(pochhammer(x+3, n)), x, k), k=0..n): seq(print(A049458_row(n)),n=0..8); # _Peter Luschny_, May 16 2013
%t t[n_, k_] := (-1)^(n - k)*Coefficient[ Pochhammer[x + 3, n], x, k]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 17 2013, after _Peter Luschny_ *)
%o (Haskell)
%o a049458 n k = a049458_tabl !! n !! k
%o a049458_row n = a049458_tabl !! n
%o a049458_tabl = map fst $ iterate (\(row, i) ->
%o (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 3)
%o -- _Reinhard Zumkeller_, Mar 11 2014
%Y Unsigned column sequences are: A001710-A001714. Row sums (signed triangle): (n+1)!*(-1)^n. Row sums (unsigned triangle): A001715(n+3).
%Y Cf. A000035 A084938.
%Y Cf. A094645, A094646.
%Y A143492, A143495. - _Peter Bala_, Aug 25 2008
%K sign,easy,tabl
%O 0,2
%A _Wolfdieter Lang_
%E Second formula corrected by _Philippe Deléham_, Nov 09 2008
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