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A051339 Generalized Stirling number triangle of first kind. 10
1, -7, 1, 56, -15, 1, -504, 191, -24, 1, 5040, -2414, 431, -34, 1, -55440, 31594, -7155, 805, -45, 1, 665280, -434568, 117454, -16815, 1345, -57, 1, -8648640, 6314664, -1961470, 336049, -34300, 2086, -70, 1, 121080960, -97053936, 33775244, -6666156, 816249, -63504, 3066, -84, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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).

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+6)*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)^7).

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.

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]

EXAMPLE

{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).

MATHEMATICA

a[n_, m_] := Pochhammer[m + 1, n - m] SeriesCoefficient[Log[1 + x]^m/(1 + x)^7, {x, 0, n}];

Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 29 2019 *)

PROG

(Haskell)

a051339 n k = a051339_tabl !! n !! k

a051339_row n = a051339_tabl !! n

a051339_tabl = map fst $ iterate (\(row, i) ->

   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 7)

-- Reinhard Zumkeller, Mar 11 2014

CROSSREFS

The first (m=0) column sequence is A001730. Row sums (signed triangle): A001725(n+5)*(-1)^n. Row sums (unsigned triangle): A049388(n).

Cf. A000035 A084938.

Sequence in context: A075502 A052104 A144450 * A134141 A237111 A281620

Adjacent sequences:  A051336 A051337 A051338 * A051340 A051341 A051342

KEYWORD

sign,easy,tabl

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified September 18 22:46 EDT 2020. Contains 337174 sequences. (Running on oeis4.)