A051379 Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 7.

1, -8, 1, 72, -17, 1, -720, 242, -27, 1, 7920, -3382, 539, -38, 1, -95040, 48504, -9850, 995, -50, 1, 1235520, -725592, 176554, -22785, 1645, -63, 1, -17297280, 11393808, -3197348, 495544, -45815, 2527, -77, 1, 259459200, -188204400, 59354028, -10630508, 1182769, -83720, 3682, -92, 1

Offset

0,2

Comments

T(n, m) = ^8P_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-(8+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(8*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+7)*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)^8).

Triangle (signed) = [ -8, -1, -9, -2, -10, -3, -11, -4, -12, ...] DELTA A000035; triangle (unsigned) = [8, 1, 9, 2, 10, 3, 11, 4, 12, 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, 8), for n=1,2,...; i=0..n. - Milan Janjic, Dec 21 2008

Example

Triangle begins:
     1;
    -8,   1;
    72, -17,   1;
  -720, 242, -27, 1;
   ...
s(2, x)=72-17*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)^8, {x, 0, n}];
Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 29 2019 *)

Prog

(Haskell)
a051379 n k = a051379_tabl !! n !! k
a051379_row n = a051379_tabl !! n
a051379_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 8)
-- Reinhard Zumkeller, Mar 12 2014

Crossrefs

The first (m=0) column sequence is: A049388. Row sums (signed triangle): A001730(n+6)*(-1)^n. Row sums (unsigned triangle): A049389(n).

Cf. A000035 A084938.

Similar generalizations: A049444 (r=1), A049458 (r=2), A049459 (r=3), A049460 (r=4), A051338 (r=5), A051339 (r=6), A051390 (r=8), A051523 (r=9).

Sequence in context: A075503 A260040 A380860 * A143499 A114152 A347111

Adjacent sequences: A051376 A051377 A051378 * A051380 A051381 A051382

Keyword

sign,easy,tabl

Author

Wolfdieter Lang

Extensions

Typo fixed in data by Reinhard Zumkeller, Mar 12 2014

Name changed by Thomas Scheuerle, Feb 04 2026

Status

approved