OFFSET
1,4
COMMENTS
Row sums = A007837.
Sum k! * T(n,k) = A032011.
Sum k * T(n,k) = A131623. - Geoffrey Critzer, Aug 30 2012.
T(n,k) is also the number of words w of length n over a k-ary alphabet {a1,a2,...,ak} with #(w,a1) > #(w,a2) > ... > #(w,ak) > 0, where #(w,x) counts the letters x in word w. T(5,2) = 15: aaaab, aaaba, aaabb, aabaa, aabab, aabba, abaaa, abaab, ababa, abbaa, baaaa, baaab, baaba, babaa, bbaaa. - Alois P. Heinz, Jun 21 2013
LINKS
Alois P. Heinz, Rows n = 1..500, flattened
FORMULA
E.g.f.: Product_{n>=1} (1+y*x^n/n!).
EXAMPLE
Triangle T(n,k)begins:
1;
1;
1, 3;
1, 4;
1, 15;
1, 21, 60;
1, 63, 105;
1, 92, 448;
1, 255, 2016;
1, 385, 4980, 12600;
1, 1023, 15675, 27720;
1, 1585, 61644, 138600;
1, 4095, 155155, 643500;
1, 6475, 482573, 4408404;
1, 16383, 1733550, 12687675, 37837800;
...
MAPLE
b:= proc(n, i, t, v) option remember; `if`(t=1, 1/(n+v)!,
add(b(n-j, j, t-1, v+1)/(j+v)!, j=i..n/t))
end:
T:= (n, k)->`if`(k*(k+1)/2>n, 0, n!*b(n-k*(k+1)/2, 0, k, 1)):
seq(seq(T(n, k), k=1..floor(sqrt(2+2*n)-1/2)), n=1..20);
# Alois P. Heinz, Jun 21 2013
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, 1, b(n, i-1)+binomial(n, i)*
expand(x*b(n-i, min(n-i, i-1)))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=1..20); # Alois P. Heinz, Sep 27 2019
MATHEMATICA
nn=10; p=Product[1+y x^i/i!, {i, 1, nn}]; Range[0, nn]! CoefficientList[ Series[p, {x, 0, nn}], {x, y}]//Grid (* Geoffrey Critzer, Aug 30 2012 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Vladeta Jovovic, Sep 04 2007
STATUS
approved