login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A051338 Generalized Stirling number triangle of first kind. 10
1, -6, 1, 42, -13, 1, -336, 146, -21, 1, 3024, -1650, 335, -30, 1, -30240, 19524, -5000, 635, -40, 1, 332640, -245004, 74524, -11985, 1075, -51, 1, -3991680, 3272688, -1139292, 218344, -24885, 1687, -63, 1, 51891840, -46536624, 18083484, -3977764, 541849, -46816, 2506, -76, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
a(n,m)= ^6P_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-(6+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(6*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
FORMULA
a(n, m)= a(n-1, m-1) - (n+5)*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)^6).
Triangle (signed) = [ -6, -1, -7, -2, -8, -3, -9, -4, -10, ...] DELTA A000035; triangle (unsigned) = [6, 1, 7, 2, 8, 3, 9, 4, 10, 5, 11, ...] 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,6), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008
EXAMPLE
{1}; {-6,1}; {42,-13,1}; {-336,146,-21,1}; ... s(2,x)= 42-13*x+x^2; S1(2,x)= -x+x^2 (Stirling1).
MATHEMATICA
t[n_, i_] = Sum[(-1)^k*Binomial[n, k]*Pochhammer[6, k]*StirlingS1[n - k, i], {k, 0, n - i}]; Flatten[Table[t[n, i], {n, 0, 8}, {i, 0, n}]][[1 ;; 45]] (* Jean-François Alcover, Jun 01 2011, after Milan Janjic *)
PROG
(Haskell)
a051338 n k = a051338_tabl !! n !! k
a051338_row n = a051338_tabl !! n
a051338_tabl = map fst $ iterate (\(row, i) ->
(zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 6)
-- Reinhard Zumkeller, Mar 11 2014
CROSSREFS
Unsigned m=0 column sequence is: A001725. Row sums (signed triangle): A001720(n+4)*(-1)^n. Row sums (unsigned triangle): A001730(n+6).
Sequence in context: A145357 A035529 A135893 * A062138 A143498 A144356
KEYWORD
sign,easy,tabl
AUTHOR
STATUS
approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 01:34 EDT 2024. Contains 370952 sequences. (Running on oeis4.)