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%I #20 Oct 29 2019 09:06:29
%S 1,-7,1,56,-15,1,-504,191,-24,1,5040,-2414,431,-34,1,-55440,31594,
%T -7155,805,-45,1,665280,-434568,117454,-16815,1345,-57,1,-8648640,
%U 6314664,-1961470,336049,-34300,2086,-70,1,121080960,-97053936,33775244,-6666156,816249,-63504,3066,-84,1
%N Generalized Stirling number triangle of first kind.
%C a(n,m)= ^7P_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-(7+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. In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer for (exp(7*t),exp(t)-1).
%H Reinhard Zumkeller, <a href="/A051339/b051339.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+6)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n<m; a(n, -1) := 0, a(0, 0)=1.
%F E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^7).
%F Triangle (signed) = [ -7, -1, -8, -2, -9, -3, -10, -4, -11, -5, ...] DELTA A000035; triangle (unsigned) = [7, 1, 8, 2, 9, 3, 10, 4, ...] DELTA A000035; 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,7), for n=1,2,...;i=0...n. [From _Milan Janjic_, Dec 21 2008]
%e {1}; {-7,1}; {56,-15,1}; {-504,191,-24,1}; ... s(2,x)= 56-15*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)^7, {x, 0, n}];
%t Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 29 2019 *)
%o (Haskell)
%o a051339 n k = a051339_tabl !! n !! k
%o a051339_row n = a051339_tabl !! n
%o a051339_tabl = map fst $ iterate (\(row, i) ->
%o (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 7)
%o -- _Reinhard Zumkeller_, Mar 11 2014
%Y The first (m=0) column sequence is A001730. Row sums (signed triangle): A001725(n+5)*(-1)^n. Row sums (unsigned triangle): A049388(n).
%Y Cf. A000035 A084938.
%K sign,easy,tabl
%O 0,2
%A _Wolfdieter Lang_
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