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Number of matchings in the n-sun graph.
3

%I #40 Dec 05 2019 15:01:03

%S 1,3,8,27,100,393,1624,7017,31558,147177,709592,3527769,18045428,

%T 94797147,510594056,2815698483,15877236898,91442860467,537363872008,

%U 3219075448251,19641501806932,121974079707225,770381455577048,4945495555291017,32249369951426822

%N Number of matchings in the n-sun graph.

%C Extended to a(0)-a(2) using the sum/recurrence. - _Eric W. Weisstein_, Oct 03 2017

%H Andrew Howroyd and Vaclav Kotesovec, <a href="/A192856/b192856.txt">Table of n, a(n) for n = 0..780</a> (terms 3..50 from Andrew Howroyd; terms a(0..2) corrected by _Georg Fischer_, Jan 20 2019)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentEdgeSet.html">Independent Edge Set</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching.html">Matching</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SunGraph.html">Sun Graph</a>

%F a(n) = A000085(n) + 2*(Sum_{ k=1..n } n * binomial(2*n-k-1, k-1) * A000085(n-k) / k). - _Andrew Howroyd_, Feb 28 2016, corrected by _Vaclav Kotesovec_, Mar 06 2016

%F Recurrence (for n>=7): (n-3)*a(n) = 3*(n-3)*a(n-1) + (n^2 - 4*n + 5)*a(n-2) - 3*(n-1)*a(n-3) + (n-1)*a(n-4). - _Vaclav Kotesovec_, Mar 06 2016

%F a(n) ~ exp(3*sqrt(n) - n/2 - 13/4) * n^(n/2) / sqrt(2) * (1 + 39/(8*sqrt(n))). - _Vaclav Kotesovec_, Mar 06 2016

%t Table[Sum[(2 j - 1)!! Binomial[n, 2 j], {j, 0, n/2}] + 2 Sum[n Binomial[2 n - k - 1, k - 1] Sum[(2 j - 1)!! Binomial[n - k, 2 j], {j, 0, (n - k)/2}]/k, {k, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 06 2016 *)

%t Join[{1}, RecurrenceTable[{(n - 3) a[n] == 3 (n - 3) a[n - 1] + (n^2 - 4 n

%t + 5) a[n - 2] - 3 (n - 1) a[n - 3] + (n - 1) a[n - 4], a[1]==3, a[2]==8, a[3]==27, a[4] == 100}, a, {n, 1, 20}]] (* _Eric W. Weisstein_, Oct 03 2017, amended by _Georg Fischer_, Dec 05 2019 *)

%Y Cf. A000085.

%K nonn

%O 0,2

%A _Eric W. Weisstein_, Jul 11 2011

%E a(7)-a(20) from _Andrew Howroyd_, Feb 28 2016

%E a(0)-a(2) from _Eric W. Weisstein_, Oct 03 2017