A361240
Number of nonequivalent noncrossing triangular cacti with n triangles up to rotation and reflection.
Original entry on oeis.org
1, 1, 1, 4, 19, 124, 931, 7801, 68685, 630850, 5966610, 57808920, 571178751, 5737672339, 58455577800, 602859484608, 6283968796705, 66119472527814, 701526880303315, 7498841163925819, 80696081185766970, 873654670250482120, 9510760874015305314, 104056578392127906720
Offset: 0
A367872
Number of dissections of a convex (4n+4)-sided polygon into n hexagons and one square (up to equivalence).
Original entry on oeis.org
1, 4, 30, 272, 2695, 28080, 302064, 3321120, 37095201, 419276660, 4782798020, 54960207120, 635339153865, 7380876649216, 86101923008160, 1007980225327680, 11836181297108565, 139353762142502100
Offset: 0
For n=0, there is just one square, so that a(0)=1. For n=1, one can dissect an octagon in 8 ways into a hexagon and a square. In this case, the equivalence relation just relates every such dissection to its half rotated image, so that a(1)=4.
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Table[Binomial[5*n + 2, n]*(n + 3)/(4*n + 3), {n, 0, 50}]
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for(n=0,25, print1(binomial(5*n+2,n)*(n+3)/(4*n+3), ", "))
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def A367872(n):
return binomial(5*n+2, n) * (n+3) / (4*n+3)
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