A051380 Generalized Stirling number triangle of first kind.

1, -9, 1, 90, -19, 1, -990, 299, -30, 1, 11880, -4578, 659, -42, 1, -154440, 71394, -13145, 1205, -55, 1, 2162160, -1153956, 255424, -30015, 1975, -69, 1, -32432400, 19471500, -4985316, 705649, -59640, 3010, -84, 1, 518918400, -343976400, 99236556, -16275700, 1659889, -107800, 4354, -100, 1

Offset: 0

Author

Wolfdieter Lang

Keywords

• sign
• easy
• tabl

Comments

a(n,m)= ^9P_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-(9+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(9*t),exp(t)-1).

Examples

			{1}; {-9,1}; {90,-19,1}; {-990,299,-30,1}; ... s(2,x)= 90-19*x+x^2; S1(2,x)= -x+x^2 (Stirling1).
		

Links

• Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
• D. S. Mitrinovic, M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).

Crossrefs

The first (m=0) column sequence is: A049389. Row sums (signed triangle): A049388(n)*(-1)^n. Row sums (unsigned triangle): A049398(n).

Cf. A000035 A084938.

Programs

Haskell

a051380 n k = a051380_tabl !! n !! k
a051380_row n = a051380_tabl !! n
a051380_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 9)
-- Reinhard Zumkeller, Mar 12 2014

Mathematica

a[n_, m_] := Pochhammer[m + 1, n - m] SeriesCoefficient[Log[1 + x]^m/(1 + x)^9, {x, 0, n}];
Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 29 2019 *)

Formula

a(n, m)= a(n-1, m-1) - (n+8)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n

E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^9).

Triangle (signed) = [ -9, -1, -10, -2, -11, -3, -12, -4, -13, ...] DELTA A000035; triangle (unsigned) = [9, 1, 10, 2, 11, 3, 12, 4, 13, 5, ...] DELTA A000035; where DELTA is Deléham's operator defined in A084938.

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,9), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

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