A185100
Dihedral unlabeled Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n unlabeled points equally spaced on a circle, up to rotations and reflections of the circle.
Original entry on oeis.org
1, 1, 2, 2, 4, 5, 11, 16, 36, 65, 150, 312, 756, 1743, 4353, 10732, 27489, 70379, 183866, 481952, 1277784, 3402661, 9126689, 24584870, 66567924, 180939737, 493801694, 1352203202, 3715137460, 10237545525, 28291018283, 78384998904, 217715672036, 606103034821, 1691020991782, 4727601528674, 13242641322252, 37162431389051, 104469244613429
Offset: 0
-
a1006[0] = 1; a1006[n_Integer] := a1006[n] = a1006[n - 1] + Sum[a1006[k]* a1006[n - 2 - k], {k, 0, n - 2}];
a142150[n_] := n*(1 + (-1)^n)/4;
a2426[n_] := Coefficient[(1 + x + x^2)^n, x, n];
a175954[0] = 1; a175954[n_] := (1/n)*(a1006[n] + a142150[n]*a1006[n/2 - 1] + Sum[EulerPhi[n/d]*a2426[d], {d, Most @Divisors[n]}]);
a5773[0] = 1; a5773[n_] := Sum[k/n*Sum[Binomial[n, j]*Binomial[j, 2*j - n - k], {j, 0, n}], {k, 1, n}];
a[0] = 1;
a[n_?OddQ] := With[{m = (n-1)/2}, (1/2)*(a175954[2*m + 1] + a5773[m + 1])];
a[n_?EvenQ] := With[{m = n/2}, (1/4)*(2*a175954[2*m] + a5773[m] + a5773[m + 1] + a1006[m - 1])];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)
A378941
Number of Motzkin paths of length n up to reversal.
Original entry on oeis.org
1, 1, 2, 3, 7, 13, 32, 70, 179, 435, 1142, 2947, 7889, 21051, 57192, 155661, 427795, 1179451, 3271214, 9102665, 25434661, 71282431, 200406472, 564905068, 1596435581, 4521772933, 12835116530, 36504093693, 104012240063, 296871993373, 848694481664, 2429882584254, 6966789756243
Offset: 0
The Motzkin paths for a(1)..a(5) are:
a(1) = 1: H;
a(2) = 2: HH, UD;
a(3) = 3: HHH, UHD, HUD=UDH;
a(4) = 7: HHHH, HUDH, UHHD, UUDD, UDUD, HHUD=UDHH, HUHD=UHDH.
a(5) = 13: HHHHH, HUHDH, UHHHD, UUHDD, UDHUD, HHHUD=UDHHH, HHUHD=UHDHH, HHUDH=HUDHH, HUHHD=UHHDH, HUUDD=UUDDH, HUDUD=UDUDH, UHUDD=UUHDD, UHDUD=UPUHD.
A303874
Number of noncrossing partitions of an n-set up to rotation with all blocks having a prime number of elements.
Original entry on oeis.org
1, 0, 1, 1, 1, 2, 3, 5, 8, 17, 37, 71, 179, 366, 919, 2069, 5027, 12053, 29098, 71846, 175485, 437438, 1087122, 2723326, 6860525, 17301606, 43957596, 111748571, 285591775, 731432424, 1879009622, 4841510973, 12500324496, 32366232373, 83962263464, 218309244314
Offset: 0
-
\\ number of partitions with restricted block sizes
NCPartitionsModCyclic(v)={ my(n=#v);
my(p=serreverse(x/(1 + sum(k=1, #v, x^k*v[k])) + O(x^2*x^n) )/x);
my(vars=variables(p));
my(varpow(r,d)=substvec(r + O(x^(n\d+1)), vars, apply(t->t^d, vars)));
my(q=x*deriv(p)/p);
my(T=sum(k=1, #v, my(t=v[k]); if(t, x^k*t*sumdiv(k, d, eulerphi(d) * varpow(p,d)^(k/d))/k)));
T + 2 + intformal(sum(d=1,n,eulerphi(d)*varpow(q,d))/x) - p
}
Vec(NCPartitionsModCyclic(vector(40, k, isprime(k))))
A175955
Number of ways to connect with nonintersecting chords n unlabeled points equally spaced on a circle such that the resulting configuration is not invariant w.r.t. rotation any angle < 2*Pi.
Original entry on oeis.org
1, 0, 1, 1, 4, 6, 18, 36, 92, 209, 527, 1269, 3218, 8063, 20701, 53209, 138634, 362789, 957857, 2541735, 6787960, 18214250, 49120018, 133024306, 361736098, 987284765, 2703991469, 7429359867, 20473889132, 56579399002, 156766505690
Offset: 1
For n=2, there are only two configurations possible: two diametrically located points on a circle connected or not connected with a chord. Since both these configurations are invariant w.r.t. rotation by angle Pi, a(2)=0.
Showing 1-4 of 4 results.
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