OFFSET
0,5
COMMENTS
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 225.
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
FORMULA
E.g.f.: exp[sinh(z)+t(cosh(z)-1)].
EXAMPLE
T(4,1) = 7 because we have 1234, 14|2|3, 1|24|3, 1|2|34, 13|2|4, 1|23|4 and 12|3|4.
Triangle starts:
1;
1;
1, 1;
2, 3;
5, 7, 3;
12, 25, 15;
37, 91, 60, 15;
...
MAPLE
G:=exp(sinh(z)+t*(cosh(z)-1)): Gser:=simplify(series(G, z=0, 16)): for n from 0 to 13 do P[n]:=sort(n!*coeff(Gser, z, n)) od: for n from 0 to 13 do seq(coeff(P[n], t, j), j=0..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
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`(irem(i, 2)=0, x^j, 1), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..15); # Alois P. Heinz, Mar 08 2015
MATHEMATICA
nn = 10; Range[0, nn]! CoefficientList[Series[Exp[y (Cosh[x] - 1) + Sinh[x]], {x, 0, nn}], {x, y}] // Grid (* Geoffrey Critzer, Aug 28 2012*)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Oct 28 2006
STATUS
approved