A049460 Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 4.

1, -5, 1, 30, -11, 1, -210, 107, -18, 1, 1680, -1066, 251, -26, 1, -15120, 11274, -3325, 485, -35, 1, 151200, -127860, 44524, -8175, 835, -45, 1, -1663200, 1557660, -617624, 134449, -17360, 1330, -56, 1, 19958400, -20355120, 8969148, -2231012, 342769, -33320, 2002, -68, 1

Offset

0,2

Comments

T(n, m) = ^5P_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-(5+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(5*t), exp(t)-1).

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+4)*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)^5).

Triangle (signed) = [ -5, -1, -6, -2, -7, -3, -8, -4, -9, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, ...]; triangle (unsigned) = [5, 1, 6, 2, 7, 3, 8, 4, 9, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...]; 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, 5), for n=1,2,...; i=0..n. - Milan Janjic, Dec 21 2008

Example

Triangle begins:
     1;
    -5,   1;
    30, -11,   1;
  -210, 107, -18, 1;
   ...
s(2, x)= 30-11*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)^5, {x, 0, n}];
Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 29 2019 *)

Prog

(Haskell)
a049460 n k = a049460_tabl !! n !! k
a049460_row n = a049460_tabl !! n
a049460_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 5)
-- Reinhard Zumkeller, Mar 11 2014

Crossrefs

Unsigned column sequences are: A001720-A001724. Row sums (signed triangle): A001715(n+3)*(-1)^n. Row sums (unsigned triangle): A001725(n+5).

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

Similar generalizations: A049444 (r=1), A049458 (r=2), A049459 (r=3), A051338 (r=5), A051339 (r=6), A051379 (r=7), A051390 (r=8), A051523 (r=9).

Sequence in context: A144890 A144891 A135892 * A145926 A062140 A144355

Adjacent sequences: A049457 A049458 A049459 * A049461 A049462 A049463

Keyword

sign,easy,tabl

Author

Wolfdieter Lang

Extensions

Second formula corrected by Philippe Deléham, Nov 10 2008

Name changed by Thomas Scheuerle, Feb 04 2026

Status

approved