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 A046860 Triangle giving a(n,k) = number of k-colored labeled graphs with n nodes. 5
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 Column #1 gives A000683. Main diagonal gives A011266. Row sums give A334282. Cf. A000683, A006201, A006202. Sequence in context: A079621 A285061 A285066 * A089505 A300083 A062328 Adjacent sequences:  A046857 A046858 A046859 * A046861 A046862 A046863 KEYWORD tabl,easy,nice,nonn AUTHOR EXTENSIONS More terms from Vladeta Jovovic, Feb 04 2000 STATUS approved

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Last modified August 1 00:13 EDT 2021. Contains 346377 sequences. (Running on oeis4.)