A049458 - OEIS

A049458

Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 2.

1, -3, 1, 12, -7, 1, -60, 47, -12, 1, 360, -342, 119, -18, 1, -2520, 2754, -1175, 245, -25, 1, 20160, -24552, 12154, -3135, 445, -33, 1, -181440, 241128, -133938, 40369, -7140, 742, -42, 1, 1814400, -2592720, 1580508, -537628

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

T(n, m) = ^3P_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-(3+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(3*t), exp(t)-1).

See A143492 for the unsigned version of this array and A143495 for the inverse. - Peter Bala, Aug 25 2008

LINKS

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

Dragoslav S. Mitrinović and Ružica S. Mitrinović, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962); alternative link.

FORMULA

T(n, m) = T(n-1, m-1) - (n+2)*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)^3).

Triangle (signed) = [ -3, -1, -4, -2, -5, -3, -6, -4, -7, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; triangle (unsigned) = [3, 1, 4, 2, 5, 3, 6, 4, 7, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deléham's operator defined in A084938 (unsigned version in A143492).

E.g.f.: (1+y)^(x-3). - Vladeta Jovovic, May 17 2004

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,3), for n=1,2,...; i=0..n. - Milan Janjic, Dec 21 2008

EXAMPLE

Triangle begins:

-3, 1;

12, -7, 1;

-60, 47, -12, 1;

360, -342, 119, -18, 1;

...

s(2, x) = 12-7*x+x^2. S1(2, x) = -x+x^2 (Stirling1 polynomial).

MAPLE

A049458_row := n -> seq((-1)^(n-k)*coeff(expand(pochhammer(x+3, n)), x, k), k=0..n): seq(print(A049458_row(n)), n=0..8); # Peter Luschny, May 16 2013

MATHEMATICA

t[n_, k_] := (-1)^(n - k)*Coefficient[ Pochhammer[x + 3, n], x, k]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 17 2013, after Peter Luschny *)

PROG

(Haskell)

a049458 n k = a049458_tabl !! n !! k

a049458_row n = a049458_tabl !! n

a049458_tabl = map fst $ iterate (\(row, i) ->

(zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 3)

-- Reinhard Zumkeller, Mar 11 2014

CROSSREFS

Unsigned column sequences are: A001710-A001714. Row sums (signed triangle): (n+1)!*(-1)^n. Row sums (unsigned triangle): A001715(n+3).

Cf. A000035, A084938, A094645, A094646, A143492, A143495.

Similar generalizations: A049444 (r=1), A049459 (r=3), A049460 (r=4), A051338 (r=5), A051339 (r=6), A051379 (r=7), A051390 (r=8), A051523 (r=9).

Sequence in context: A329433 A258245 A133366 * A143492 A375046 A243662