Original entry on oeis.org
-1, 1, 3, 20, 160, 1727, 22341, 337947, 5799881, 111180832, 2352448424, 54449597409, 1368516031855, 37118127188225, 1080644471447419, 33614180067524196, 1112586937337720904, 39043623554061199807, 1448021297870473796645, 56592256120004219495755, 2324706946641972649074513
Offset: 1
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f(n) = sum(k=0, n, (2*n-k)! / (k! * (n-k)!) * (-1/2)^(n-k) ); \\ A000806
lista(nn) = {my(va = vector(nn)); va[1] = 1; va[2] = 0; va[3] = 1; va[4] = 3; va[5] = 12; for (n=5, nn-1, va[n+1] = 2*va[n] + (2*n-3)*va[n-1] - (2*n-5)*va[n-2] + 2*va[n-3] - va[n-4];); my(w=vector(nn-1, n, (va[n] + abs(f(n-1)))/2)); vector(#w-1, k, w[k+1] - w[k]);} \\ Michel Marcus, Jul 28 2020
A271218
Number of symmetric linked diagrams with n links and no simple link.
Original entry on oeis.org
1, 0, 1, 3, 12, 39, 167, 660, 3083, 13961, 70728, 355457, 1936449, 10587960, 61539129, 361182139, 2224641540, 13880534119, 90090083047, 593246514588, 4038095508691, 27905008440273, 198401618299920, 1432253086621377, 10600146578310209, 79639887325700592, 611739960145556273
Offset: 0
For n=0 the a(0)=1 solution is { ∅ }.
For n=1 there are no solutions since the link in a diagram with one link, 11, is simple.
For n=2 the a(2)=1 solution is { 1212 }.
For n=3 the a(3)=3 solutions are { 123123, 121323, 123231 }.
For n=4 the a(4)=12 solutions are { 12123434, 12132434, 12324341, 12314234, 12341234, 12342341, 12314324, 12341324, 12343412, 12343421, 12324143, 12342143 }.
- J. Burns, Counting a Class of Signed Permutations and Chord Diagrams related to DNA Rearrangement, Preprint.
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RecurrenceTable[{a[n]==2a[n-1]+(2n-3)a[n-2]-(2n-5)a[n-3]+2a[n-4]-a[n-5],a[0]==1,a[1]==0,a[2]==1,a[3]==3,a[4]==12},a[n],{n,20}]
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lista(nn) = {my(va = vector(nn)); va[1] = 1; va[2] = 0; va[3] = 1; va[4] = 3; va[5] = 12; for (n=5, nn-1, va[n+1] = 2*va[n] + (2*n-3)*va[n-1] - (2*n-5)*va[n-2] + 2*va[n-3] - va[n-4];); va;} \\ Michel Marcus, Jul 28 2020
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