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A046860 Triangle giving a(n,k) = number of k-colored labeled graphs with n nodes.

%I A046860 #36 May 10 2025 23:14:48
%S A046860 1,1,4,1,24,48,1,160,1152,1536,1,1440,30720,122880,122880,1,18304,
%T A046860 1152000,10813440,29491200,23592960,1,330624,65630208,1348730880,
%U A046860 7707033600,15854469120,10569646080,1,8488960,5858721792,261070258176,2853804441600,11499774935040,18940805775360,10823317585920
%N A046860 Triangle giving a(n,k) = number of k-colored labeled graphs with n nodes.
%H A046860 Alois P. Heinz, <a href="/A046860/b046860.txt">Rows n = 1..50, flattened</a>
%H A046860 R. C. Read, <a href="http://cms.math.ca/10.4153/CJM-1960-035-0">The number of k-colored graphs on labelled nodes</a>, Canad. J. Math., 12 (1960), 410-414.
%F A046860 a(n, k) = Sum_{r=1..n-1} C(n, r) 2^(r*(n-r)) a(r, k-1).
%F A046860 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
%e A046860 Triangle begins:
%e A046860   1;
%e A046860   1,     4;
%e A046860   1,    24,      48;
%e A046860   1,   160,    1152,     1536;
%e A046860   1,  1440,   30720,   122880,   122880;
%e A046860   1, 18304, 1152000, 10813440, 29491200, 23592960;
%e A046860   ...
%p A046860 a:= proc(n, k) option remember; `if`([n, k]=[0$2], 1,
%p A046860       add(binomial(n, r)*2^(r*(n-r))*a(r, k-1), r=0..n-1))
%p A046860     end:
%p A046860 seq(seq(a(n,k), k=1..n), n=1..8);  # _Alois P. Heinz_, Apr 21 2020
%t A046860 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 *)
%Y A046860 Column #1 gives A000683.
%Y A046860 Main diagonal gives A011266.
%Y A046860 Row sums give A334282.
%Y A046860 Cf. A000683, A006201, A006202.
%K A046860 tabl,easy,nice,nonn
%O A046860 1,3
%A A046860 _N. J. A. Sloane_
%E A046860 More terms from _Vladeta Jovovic_, Feb 04 2000