OFFSET
1,2
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
LINKS
Seiichi Manyama, Rows n = 1..140, flattened (Rows n = 1..60 from Vincenzo Librandi)
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
EXAMPLE
Triangle begins:
1;
-2, 1;
7, -6, 1;
-35, 40, -12, 1;
228, -315, 130, -20, 1;
-1834, 2908, -1485, 320, -30, 1;
...
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 *)
PROG
(PARI) T(n, k) = sum(j=0, n, stirling(n, j, 1)*stirling(j, k, 1)); \\ Seiichi Manyama, Feb 13 2022
CROSSREFS
KEYWORD
AUTHOR
Christian G. Bower, Feb 15 1999
STATUS
approved