OFFSET
1,3
LINKS
Alois P. Heinz, Rows n = 1..50, flattened
R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410—414.
FORMULA
a(n, k) = Sum_{r=1..n-1} C(n, r) 2^(r*(n-r)) a(r, k-1).
1 + Sum_{n>=1} Sum_{k=1..n} a(n,k)*y^k*x^n/(n!*2^C(n,2)) = 1/(1-y(E(x)-1)) where E(x) = Sum_{n>=0} x^n/(n!*2^C(n,2)). - Geoffrey Critzer, May 06 2020
EXAMPLE
Triangle begins:
1;
1, 4;
1, 24, 48;
1, 160, 1152, 1536;
1, 1440, 30720, 122880, 122880;
1, 18304, 1152000, 10813440, 29491200, 23592960;
...
MAPLE
a:= proc(n, k) option remember; `if`([n, k]=[0$2], 1,
add(binomial(n, r)*2^(r*(n-r))*a(r, k-1), r=0..n-1))
end:
seq(seq(a(n, k), k=1..n), n=1..8); # Alois P. Heinz, Apr 21 2020
MATHEMATICA
a[n_ /; n >= 1, k_ /; k >= 1] := a[n, k] = Sum[ Binomial[n, r]*2^(r*(n - r))*a[r, k - 1], {r, 1, n - 1}]; a[_, 0] = 1; Flatten[ Table[ a[n, k], {n, 1, 8}, {k, 0, n - 1}]] (* Jean-François Alcover, Dec 12 2011, after formula *)
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Feb 04 2000
STATUS
approved