A048176 Triangle read by rows: T(n, m) = S1(n, m)*10^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).

1, -10, 1, 200, -30, 1, -6000, 1100, -60, 1, 240000, -50000, 3500, -100, 1, -12000000, 2740000, -225000, 8500, -150, 1, 720000000, -176400000, 16240000, -735000, 17500, -210, 1, -50400000000, 13068000000, -1313200000, 67690000, -1960000, 32200, -280, 1, 4032000000000, -1095840000000

Offset

1,2

Comments

T(n, m) = R_n^m(a=0, b=10) in the notation of the given reference.

T(n, m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n, x) = sum(T(n, m)*x^m, m=1..n) = product(x-10*j, j=0..n-1), n >= 1, E(0, x) = 1, are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

Also the Bell transform of the sequence (-1)^n*A051262(n) without column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

Links

Table of n, a(n) for n=1..38.

D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77.

Formula

T(n, m) = T(n-1, m-1) - 10*(n-1)*T(n-1, m), n >= m >= 1; T(n, m) = 0, n < m; T(n, 0) = 0, T(1, 1) = 1. E.g.f. for m-th column of signed triangle: (((log(1+10*x))/10)^m)/m!.

Example

Triangle begins:
  {1};
  {-10,1};
  {200,-30,1};
  {-6000,1100,-60,1};
  ...
E(3,x) = 200*x-30*x^2+x^3.

Maple

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> (-1)^n*n!*10^n, 9); # Peter Luschny, Jan 28 2016

Mathematica

rows = 9;
t = Table[(-1)^n*n!*10^n, {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

Crossrefs

First (m=1) (unsigned) column sequence is: A051262(n-1). Row sums (signed triangle): A049212(n-1)*(-1)^(n-1). Row sums (unsigned triangle): A045757(n).

Similar generalizations of Stirling numbers: A008275 (Stirling1 triangle, b=1), A039683 (b=2), A051141 (b=3), A051142 (b=4), A051150 (b=5), A051151 (b=6), A051186 (b=7), A051187 (b=8), A051231 (b=9).

Sequence in context: A308282 A223512 A131367 * A127616 A277394 A191549

Adjacent sequences: A048173 A048174 A048175 * A048177 A048178 A048179

Keyword

sign,easy,tabl

Author

Wolfdieter Lang

Extensions

Name changed by Thomas Scheuerle, Feb 05 2026

Status

approved