A046860 Triangle giving a(n,k) = number of k-colored labeled graphs with n nodes.
1, 1, 4, 1, 24, 48, 1, 160, 1152, 1536, 1, 1440, 30720, 122880, 122880, 1, 18304, 1152000, 10813440, 29491200, 23592960, 1, 330624, 65630208, 1348730880, 7707033600, 15854469120, 10569646080, 1, 8488960, 5858721792, 261070258176, 2853804441600, 11499774935040, 18940805775360, 10823317585920
Offset: 1
Examples
Triangle begins: 1; 1, 4; 1, 24, 48; 1, 160, 1152, 1536; 1, 1440, 30720, 122880, 122880; 1, 18304, 1152000, 10813440, 29491200, 23592960; ...
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.
Crossrefs
Programs
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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 *)
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
Extensions
More terms from Vladeta Jovovic, Feb 04 2000