cp's OEIS Frontend

A192856 Number of matchings in the n-sun graph.

%I A192856 #41 Feb 16 2025 08:33:15
%S A192856 1,3,8,27,100,393,1624,7017,31558,147177,709592,3527769,18045428,
%T A192856 94797147,510594056,2815698483,15877236898,91442860467,537363872008,
%U A192856 3219075448251,19641501806932,121974079707225,770381455577048,4945495555291017,32249369951426822
%N A192856 Number of matchings in the n-sun graph.
%C A192856 Extended to a(0)-a(2) using the sum/recurrence. - _Eric W. Weisstein_, Oct 03 2017
%H A192856 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 A192856 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependentEdgeSet.html">Independent Edge Set</a>
%H A192856 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Matching.html">Matching</a>
%H A192856 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SunGraph.html">Sun Graph</a>
%F A192856 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 A192856 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 A192856 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 A192856 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 A192856 Join[{1}, RecurrenceTable[{(n - 3) a[n] == 3 (n - 3) a[n - 1] + (n^2 - 4 n
%t A192856 + 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 A192856 Cf. A000085.
%K A192856 nonn
%O A192856 0,2
%A A192856 _Eric W. Weisstein_, Jul 11 2011
%E A192856 a(7)-a(20) from _Andrew Howroyd_, Feb 28 2016
%E A192856 a(0)-a(2) from _Eric W. Weisstein_, Oct 03 2017