A365694
G.f. satisfies A(x) = 1 + x^3*A(x)^2 / (1 - x*A(x)).
Original entry on oeis.org
1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 42, 84, 170, 354, 740, 1549, 3269, 6945, 14811, 31711, 68177, 147091, 318313, 690837, 1503351, 3279445, 7169907, 15708485, 34482475, 75830981, 167042763, 368548926, 814341362, 1801867812, 3992172298, 8855912464, 19668236110
Offset: 0
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CoefficientList[Series[2/(1 + x + Sqrt[1 + x*(-2 + x - 4*x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 16 2023 *)
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a(n) = sum(k=0, n\3, binomial(n-2*k-1, n-3*k)*binomial(n-k+1, k)/(n-k+1));
A372413
Coefficient of x^n in the expansion of ( (1-x+x^3) / (1-x) )^n.
Original entry on oeis.org
1, 0, 0, 3, 4, 5, 21, 49, 92, 237, 595, 1331, 3169, 7787, 18487, 44108, 107036, 258349, 622371, 1508239, 3658679, 8869465, 21543005, 52399612, 127497281, 310487855, 756858661, 1846060464, 4505442967, 11003284052, 26887642756, 65735882819, 160795695676
Offset: 0
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a(n, s=3, t=1, u=1) = sum(k=0, n\s, binomial(t*n, k)*binomial((u-t+1)*n-(s-1)*k-1, n-s*k));
A352900
a(n) is the number of different ways to partition the set of vertices of a convex n-gon into intersecting polygons.
Original entry on oeis.org
0, 0, 0, 7, 28, 79, 460, 2486, 11209, 59787, 361777, 2167635, 13577211, 91919186, 650059294, 4761980740, 36508824672, 292116858616, 2424047807182, 20847409357919, 185754041370693, 1711253802075941, 16272637412753211, 159561718074359537, 1611599794862761838, 16747401536440092104
Offset: 3
For n=6, there are a(6) = 7 intersecting partitions of the convex hexagon. On vertices 1..6, they are the following pairs of triangles:
{1,3,4}, {5,6,2}
{4,5,1}, {2,3,6}
{3,4,6}, {1,2,5}
{2,3,5}, {1,4,6}
{1,2,4}, {5,6,3}
{1,6,3}, {5,4,2}
{1,3,5}, {2,4,6}
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T2(n,k) = if (n<3, 0, if (k==1, 1, k*T2(n-1,k) + binomial(n-1,2)*T2(n-3,k-1))); \\ A059022
a5(n) = if (n<3, n==0, sum(k=1, n\3, T2(n,k))); \\ A006505
a7(n) = sum(k=ceil((n+3)/2), n, (1/(n+1) * binomial(n+1, k) * binomial(2*k-n-3, n-k)) ); \\ A114997
a(n) = a5(n) - a7(n); \\ Michel Marcus, Apr 09 2022