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A051380 Generalized Stirling number triangle of first kind. 8
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

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).

FORMULA

a(n, m)= a(n-1, m-1) - (n+8)*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)^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

EXAMPLE

{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).

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 *)

PROG

(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

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.

Sequence in context: A075504 A138342 A101678 * A136238 A113394 A243754

Adjacent sequences:  A051377 A051378 A051379 * A051381 A051382 A051383

KEYWORD

sign,easy,tabl

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified May 24 18:33 EDT 2020. Contains 334578 sequences. (Running on oeis4.)