OFFSET
0,2
COMMENTS
T(n, m) = ^9P_n^m in the notation of the given reference with T(0, 0) = 1.
The monic row polynomials s(n, x) = Sum_{m=0..n} T(n, m)*x^m which are s(n, x) = Product_{k=0..n-1} x-(9+k), n >= 1 and s(0, x) = 1 satisfy s(n, x+y) = Sum_{k=0..n} binomial(n, k)*s(k, x)*S1(n-k, y), with the Stirling1 polynomials S1(n, x) = Sum_{m=1..n} A008275(n, m)*x^m 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 and 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
T(n, m) = T(n-1, m-1) - (n+8)*T(n-1, m), n >= m >= 0; T(n, m) = 0, n < m; T(n, -1) = 0, T(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_{k=0..n-i} binomial(n,k)*stirling1(n-k,i)*Product_{j=0..k-1}(-a-j), then T(n, i) = f(n, i, 9), for n=1,2,...; i=0..n. - Milan Janjic, Dec 21 2008
EXAMPLE
The triangle begins:
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
KEYWORD
AUTHOR
EXTENSIONS
Name changed by Thomas Scheuerle, Feb 06 2026
STATUS
approved