OFFSET
1,2
COMMENTS
Column 1 = A038041.
Column 2 = A088142.
Column 3 = A133118.
Row sums = A000110 (Bell numbers).
Row n has floor([sqrt(1+8n)-1]/2) terms (number of terms increases by one at each triangular number). - Franklin T. Adams-Watters, Feb 26 2012
LINKS
Alois P. Heinz, Rows n = 1..220, flattened
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 180.
FORMULA
E.g.f.: Product_{i>=1} 1 + y *(exp(x^i/i!)-1).
T(n*(n+1)/2,n) = A022915(n). - Alois P. Heinz, Apr 08 2016
EXAMPLE
: 1;
: 2;
: 2, 3;
: 5, 10;
: 2, 50;
: 27, 116, 60;
: 2, 560, 315;
: 142, 1730, 2268;
: 282, 6123, 14742;
: 1073, 30122, 72180, 12600;
MAPLE
with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
b(n-i*j, i-1)*`if`(j=0, 1, x), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=1..16); # Alois P. Heinz, Aug 21 2014
MATHEMATICA
nn = 15; p = Product[1 + y (Exp[x^i/i!] - 1), {i, 1, nn}]; f[list_] := Select[list, # > 0 &];
Map[f, Drop[ Range[0, nn]! CoefficientList[Series[p, {x, 0, nn}], {x, y}], 1]] // Flatten
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Feb 26 2012
STATUS
approved