A109345 - OEIS

%I #39 Sep 08 2022 08:45:19

%S 1,1,5,125,15625,9765625,30517578125,476837158203125,

%T 37252902984619140625,14551915228366851806640625,

%U 28421709430404007434844970703125

%N a(n) = 5^((n^2 - n)/2).

%C Sequence given by the Hankel transform (see A001906 for definition) of A078009 = {1, 1, 6, 41, 306, 2426, 20076, 171481, ...}; example: det([1, 1, 6, 41; 1, 6, 41, 306; 6, 41, 306, 2426; 41, 306, 2426, 20076]) = 5^6 = 15625.

%C a(n) is the number of simple labeled graphs, with bi-directional and non-directed edges allowed and not regarded as equivalent, on n labeled nodes. - _Mark Stander_, Feb 07 2019

%H G. C. Greubel, <a href="/A109345/b109345.txt">Table of n, a(n) for n = 0..50</a>

%F a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(5i, j).

%F G.f. A(x) satisfies: A(x) = 1 + x * A(5*x). - _Ilya Gutkovskiy_, Jun 04 2020

%p seq(5^(binomial(2+n,n)), n=-2..8); # _Zerinvary Lajos_, Jun 12 2007

%t 5^Binomial[Range[0, 12], 2] (* _G. C. Greubel_, Feb 09 2019 *)

%o (PARI) a(n)=5^binomial(n,2) \\ _Charles R Greathouse IV_, Jan 11 2012

%o (Magma) [5^Binomial(n,2): n in [0..12]]; // _G. C. Greubel_, Feb 09 2019

%o (Sage) [5^binomial(n,2) for n in (0..12)] # _G. C. Greubel_, Feb 09 2019

%o (GAP) List([0..12], n -> 5^Binomial(n,2)); # _G. C. Greubel_, Feb 09 2019

%Y Cf. A006125 (number of graphs on n labeled nodes), A047656 (number of semi-complete digraphs on n labeled nodes), A053763 (number of simple digraphs on n labeled nodes), A053764.

%K nonn,easy

%O 0,3

%A _Philippe Deléham_, Aug 21 2005