A051338 Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 5.

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

Offset

0,2

Comments

T(n, m) = ^6P_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-(6+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(6*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+5)*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)^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_{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, 6), for n=1,2,...; i=0..n. - Milan Janjic, Dec 21 2008

Example

Triangle begins:
     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).

Cf. A000035 A084938.

Similar generalizations: A049444 (r=1), A049458 (r=2), A049459 (r=3), A049460 (r=4), A051339 (r=6), A051379 (r=7), A051390 (r=8), A051523 (r=9).

Sequence in context: A145357 A035529 A135893 * A062138 A143498 A144356

Adjacent sequences: A051335 A051336 A051337 * A051339 A051340 A051341

Keyword

sign,easy,tabl

Author

Wolfdieter Lang

Extensions

Name changed by Thomas Scheuerle, Feb 04 2026

Status

approved