A039814 Matrix square of Stirling-1 triangle A008275.

1, -2, 1, 7, -6, 1, -35, 40, -12, 1, 228, -315, 130, -20, 1, -1834, 2908, -1485, 320, -30, 1, 17582, -30989, 18508, -5005, 665, -42, 1, -195866, 375611, -253400, 81088, -13650, 1232, -56, 1, 2487832, -5112570, 3805723, -1389612, 279048, -32130, 2100, -72, 1

Offset: 1

Author

Christian G. Bower, Feb 15 1999

Keywords

• sign
• tabl
• nice

Comments

Exponential Riordan array [1/((1 + x)*(1 + log(1 + x))), log(1 + log(1 + x))]. The row sums of the unsigned array give A007840 (apart from the initial term). - Peter Bala, Jul 22 2014

Also the Bell transform of (-1)^n*A003713(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

Examples

			Triangle begins:
      1;
     -2,    1;
      7,   -6,     1;
    -35,   40,   -12,   1;
    228, -315,   130, -20,   1;
  -1834, 2908, -1485, 320, -30, 1;
...
		

Links

• Seiichi Manyama, Rows n = 1..140, flattened (Rows n = 1..60 from Vincenzo Librandi)

Crossrefs

Column k=1..3 give (-1)^(n-1) * A003713(n), (-1)^n * A341587(n), (-1)^(n-1) * A341588(n).

Cf. A007840.

Cf. A039810, A039815, A039816, A039817.

Programs

Maple

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (-1)^n*add(k!*abs(Stirling1(n+1,k+1)), k=0..n), 10); # Peter Luschny, Jan 28 2016

Mathematica

max = 9; t = Table[StirlingS1[n, k], {n, 1, max}, {k, 1, max}]; t2 = t.t; Table[t2[[n, k]], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 01 2013 *)
rows = 9;
t = Table[(-1)^n*Sum[k!*Abs[StirlingS1[n+1, k+1]], {k,0,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 *)

PARI

T(n, k) = sum(j=0, n, stirling(n, j, 1)*stirling(j, k, 1)); \\ Seiichi Manyama, Feb 13 2022

Formula

E.g.f. of k-th column: ((log(1+log(1+x)))^k)/k!.

E.g.f.: 1/(1 + t)*( 1 + log(1 + t) )^(x-1) = 1 + (-2 + x)*t + (7 - 6*x + x^2)*t^2/2! + .... - Peter Bala, Jul 22 2014

T(n,k) = Sum_{j=0..n} Stirling1(n,j) * Stirling1(j,k). - Seiichi Manyama, Feb 13 2022