A109345 a(n) = 5^((n^2 - n)/2).

1, 1, 5, 125, 15625, 9765625, 30517578125, 476837158203125, 37252902984619140625, 14551915228366851806640625, 28421709430404007434844970703125

Offset: 0

Author

Philippe Deléham, Aug 21 2005

Keywords

• nonn
• easy

Comments

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.

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..50

Crossrefs

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.

Programs

GAP

List([0..12], n -> 5^Binomial(n,2)); # G. C. Greubel, Feb 09 2019

Magma

[5^Binomial(n,2): n in [0..12]]; // G. C. Greubel, Feb 09 2019

Maple

seq(5^(binomial(2+n,n)), n=-2..8); # Zerinvary Lajos, Jun 12 2007

Mathematica

5^Binomial[Range[0, 12], 2] (* G. C. Greubel, Feb 09 2019 *)

PARI

a(n)=5^binomial(n,2) \\ Charles R Greathouse IV, Jan 11 2012

Sage

[5^binomial(n,2) for n in (0..12)] # G. C. Greubel, Feb 09 2019

Formula

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

G.f. A(x) satisfies: A(x) = 1 + x * A(5*x). - Ilya Gutkovskiy, Jun 04 2020