login
This site is supported by donations to The OEIS Foundation.

 

Logo

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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).

REFERENCES

Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

LINKS

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

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).

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 17 22:51 EST 2019. Contains 319251 sequences. (Running on oeis4.)