A303048 Number of total dominating sets in the n-triangular (Johnson) graph.
0, 4, 54, 918, 31232, 2059624, 266734812, 68574627036, 35160753222400, 36021330363615408, 73782362964470935112, 302225854825997535378632, 2475866675779140063716682240, 40564755806137338166417907530592, 1329227401912999475682655581004557840
Offset: 2
Keywords
Links
- Andrew Howroyd, Table of n, a(n) for n = 2..50
- Eric Weisstein's World of Mathematics, Johnson Graph
- Eric Weisstein's World of Mathematics, Total Dominating Set
- Eric Weisstein's World of Mathematics, Triangular Graph
Programs
-
Mathematica
b[n_] := Sum[(-1)^(n - k) Binomial[n, k] 2^Binomial[k, 2], {k, 0, n}] Table[Sum[(-1)^k Binomial[n, 2 k] (2 k)!/(2^k k!) (b[n - 2 k] + (n - 2 k) b[n - 2 k - 1]), {k, 0, Floor[n/2]}], {n, 2, 20}] -
PARI
\\ here b(n) is A006129 b(n)=sum(k=0, n, (-1)^(n-k)*binomial(n, k)*2^binomial(k, 2)); a(n)=sum(k=0, n\2, (-1)^k*binomial(n,2*k)*(2*k)!/(2^k*k!)*(b(n-2*k) + (n-2*k)*b(n-2*k-1))); \\ Andrew Howroyd, Apr 20 2018
Formula
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n,2*k)*(2*k-1)!!*A290847(n-2*k). - Andrew Howroyd, Apr 20 2018
Extensions
a(8)-a(16) from Andrew Howroyd, Apr 20 2018