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%I #23 Oct 29 2019 09:06:20
%S 1,-5,1,30,-11,1,-210,107,-18,1,1680,-1066,251,-26,1,-15120,11274,
%T -3325,485,-35,1,151200,-127860,44524,-8175,835,-45,1,-1663200,
%U 1557660,-617624,134449,-17360,1330,-56,1,19958400,-20355120,8969148,-2231012,342769,-33320,2002,-68,1
%N Generalized Stirling number triangle of first kind.
%C a(n,m)= ^5P_n^m in the notation of the given reference with a(0,0) := 1.
%C The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(5+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 for (exp(5*t),exp(t)-1).
%H Reinhard Zumkeller, <a href="/A049460/b049460.txt">Rows n = 0..125 of triangle, flattened</a>
%H D. S. Mitrinovic, M. S. Mitrinovic, <a href="http://pefmath2.etf.rs/files/47/77.pdf">Tableaux d'une classe de nombres reliés aux nombres de Stirling</a>, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).
%F a(n, m)= a(n-1, m-1) - (n+4)*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)^5).
%F Triangle (signed) = [ -5, -1, -6, -2, -7, -3, -8, -4, -9, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, ...]; triangle (unsigned) = [5, 1, 6, 2, 7, 3, 8, 4, 9, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deléham's operator defined in A084938.
%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,5), for n=1,2,...;i=0...n. - _Milan Janjic_, Dec 21 2008
%e {1}; {-5,1}; {30,-11,1}; {-210,107,-18,1}; ... s(2,x)= 30-11*x+x^2; S1(2,x)= -x+x^2 (Stirling1).
%t a[n_, m_] := Pochhammer[m+1, n-m] SeriesCoefficient[Log[1+x]^m/(1+x)^5, {x, 0, n}];
%t Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 29 2019 *)
%o (Haskell)
%o a049460 n k = a049460_tabl !! n !! k
%o a049460_row n = a049460_tabl !! n
%o a049460_tabl = map fst $ iterate (\(row, i) ->
%o (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 5)
%o -- _Reinhard Zumkeller_, Mar 11 2014
%Y Unsigned column sequences are: A001720-A001724. Row sums (signed triangle): A001715(n+3)*(-1)^n. Row sums (unsigned triangle): A001725(n+5).
%Y Cf. A000035 A084938.
%K sign,easy,tabl
%O 0,2
%A _Wolfdieter Lang_
%E Second formula corrected by _Philippe Deléham_, Nov 10 2008
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