A051379 Generalized Stirling number triangle of first kind.

1, -8, 1, 72, -17, 1, -720, 242, -27, 1, 7920, -3382, 539, -38, 1, -95040, 48504, -9850, 995, -50, 1, 1235520, -725592, 176554, -22785, 1645, -63, 1, -17297280, 11393808, -3197348, 495544, -45815, 2527, -77, 1, 259459200, -188204400, 59354028, -10630508, 1182769, -83720, 3682, -92, 1

Offset: 0

Author

Wolfdieter Lang

Keywords

• sign
• easy
• tabl

Comments

a(n,m)= ^8P_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-(8+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(8*t),exp(t)-1).

Examples

			{1}; {-8,1}; {72,-17,1}; {-720,242,-27,1}; ... s(2,x)=72-17*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: A049388. Row sums (signed triangle): A001730(n+6)*(-1)^n. Row sums (unsigned triangle): A049389(n).

Cf. A000035 A084938.

Programs

Haskell

a051379 n k = a051379_tabl !! n !! k
a051379_row n = a051379_tabl !! n
a051379_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 8)
-- Reinhard Zumkeller, Mar 12 2014

Mathematica

a[n_, m_] := Pochhammer[m + 1, n - m] SeriesCoefficient[Log[1 + x]^m/(1 + x)^8, {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+7)*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)^8).

Triangle (signed) = [ -8, -1, -9, -2, -10, -3, -11, -4, -12, ...] DELTA A000035; triangle (unsigned) = [8, 1, 9, 2, 10, 3, 11, 4, 12, 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,8), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

Extensions

Typo fixed in data by Reinhard Zumkeller, Mar 12 2014

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