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A234618 Numbers of undirected cycles in the n-crown graph.

%I A234618 #33 Feb 16 2025 08:33:21
%S A234618 1,28,586,16676,674171,36729512,2591431284,229610080632,
%T A234618 24945009633237,3259554588092452,504229440385599358,
%U A234618 91120169013941688700,19019291896651737256463,4540685283391286195445008,1229402290052883559000280168,374675876836087520170128786864
%N A234618 Numbers of undirected cycles in the n-crown graph.
%H A234618 Andrew Howroyd, <a href="/A234618/b234618.txt">Table of n, a(n) for n = 3..50</a>
%H A234618 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CrownGraph.html">Crown Graph</a>
%H A234618 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%F A234618 a(n) = Sum_{k=2..n} binomial(n,k) * ( (-1)^k*(k-1)! + Sum_{j=0..k} Sum_{i=0..k-1} (-1)^i*i!*(k-i)!*(k-i-1)!*binomial(k,k-j)*binomial(n-k,j)*binomial(k-j,i)*binomial(2*k-i-1,i)/2 ). - _Andrew Howroyd_, Feb 24 2016
%F A234618 Recurrence: (n-3)*(180*n^5 - 3462*n^4 + 25685*n^3 - 91106*n^2 + 152414*n - 93847)*a(n) = (360*n^8 - 8904*n^7 + 93172*n^6 - 538135*n^5 + 1875502*n^4 - 4041070*n^3 + 5268157*n^2 - 3817934*n + 1189124)*a(n-1) - (n-1)*(180*n^9 - 5262*n^8 + 67445*n^7 - 497202*n^6 + 2321291*n^5 - 7107149*n^4 + 14233985*n^3 - 17904305*n^2 + 12741400*n - 3858611)*a(n-2) - (n-2)*(n-1)*(180*n^9 - 5442*n^8 + 71807*n^7 - 543239*n^6 + 2598146*n^5 - 8144697*n^4 + 16705322*n^3 - 21515171*n^2 + 15619923*n - 4754598)*a(n-3) + 2*(n-3)*(n-2)*(n-1)*(540*n^7 - 12585*n^6 + 122039*n^5 - 636205*n^4 + 1920840*n^3 - 3360924*n^2 + 3186108*n - 1302080)*a(n-4) + 2*(n-4)*(n-3)*(n-2)*(n-1)*(540*n^7 - 13806*n^6 + 145494*n^5 - 814365*n^4 + 2591726*n^3 - 4628556*n^2 + 4207415*n - 1449736)*a(n-5) - (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(1440*n^6 - 28956*n^5 + 230284*n^4 - 915485*n^3 + 1878786*n^2 - 1811640*n + 577483)*a(n-6) + 3*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(180*n^5 - 2562*n^4 + 13637*n^3 - 33023*n^2 + 34309*n - 10136)*a(n-7). - _Vaclav Kotesovec_, Feb 25 2016
%F A234618 a(n) ~ Pi * BesselI(0,2) * n^(2*n) / exp(2*n+2). - _Vaclav Kotesovec_, Feb 25 2016
%t A234618 a[n_] := Sum[Binomial[n, k]*((-1)^k*(k - 1)! + Sum[Sum[(-1)^i*i!*(k - i)!*(k - i - 1)!*Binomial[k, k - j]*Binomial[n - k, j]*Binomial[k - j, i]*Binomial[2*k - i - 1, i]/2, {i, 0, k - 1}], {j, 0, k}]), {k, 2, n}];
%t A234618 Table[a[n], {n, 3, 18}] (* _Jean-François Alcover_, Oct 02 2017, after _Andrew Howroyd_ *)
%t A234618 RecurrenceTable[{(n - 3) (180 n^5 - 3462 n^4 + 25685 n^3 - 91106 n^2 + 152414 n - 93847) a[n] == (360 n^8 - 8904 n^7 + 93172 n^6 - 538135 n^5 + 1875502 n^4 - 4041070 n^3 + 5268157 n^2 - 3817934 n + 1189124) a[n - 1] - (n - 1) (180 n^9 - 5262 n^8 + 67445 n^7 - 497202 n^6 + 2321291 n^5 - 7107149 n^4 + 14233985 n^3 - 17904305 n^2 + 12741400 n - 3858611) a[n - 2] - (n - 2) (n - 1) (180 n^9 - 5442 n^8 + 71807 n^7 - 543239 n^6 + 2598146 n^5 - 8144697 n^4 + 16705322 n^3 - 21515171 n^2 + 15619923 n - 4754598) a[n - 3] + 2 (n - 3) (n - 2) (n - 1) (540 n^7 - 12585 n^6 + 122039 n^5 - 636205 n^4 + 1920840 n^3 - 3360924 n^2 + 3186108 n - 1302080) a[n - 4] + 2 (n - 4) (n - 3) (n - 2) (n - 1) (540 n^7 - 13806 n^6 + 145494 n^5 - 814365 n^4 + 2591726 n^3 - 4628556 n^2 + 4207415 n - 1449736) a[n - 5] - (n - 5) (n - 4) (n - 3) (n - 2) (n - 1) (1440 n^6 - 28956 n^5 + 230284 n^4 - 915485 n^3 + 1878786 n^2 - 1811640 n + 577483) a[n - 6] + 3 (n - 6) (n - 5) (n - 4) (n - 3) (n - 2) (n - 1) (180 n^5 - 2562 n^4 + 13637 n^3 - 33023 n^2 + 34309 n - 10136) a[n - 7], a[3] == 1, a[4] == 28, a[5] == 586, a[6] == 16676, a[7] == 674171, a[8] == 36729512, a[9] == 2591431284}, a, {n, 3, 20}] (* _Eric W. Weisstein_, Oct 02 2017 *)
%Y A234618 Cf. A070910, A094047, A137886.
%K A234618 nonn
%O A234618 3,2
%A A234618 _Eric W. Weisstein_, Dec 28 2013
%E A234618 a(13) from _Eric W. Weisstein_, Jan 08 2014
%E A234618 a(14) from _Eric W. Weisstein_, Apr 09 2014
%E A234618 a(15)-a(16) from _Andrew Howroyd_, Feb 24 2016