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Triangle read by rows: T(n,k) = number of nonequivalent dissections of an n-gon into k polygons by nonintersecting diagonals up to rotation and reflection.
+10
8
1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 1, 2, 6, 7, 4, 1, 3, 11, 24, 24, 12, 1, 3, 17, 51, 89, 74, 27, 1, 4, 26, 109, 265, 371, 259, 82, 1, 4, 36, 194, 660, 1291, 1478, 891, 228, 1, 5, 50, 345, 1477, 3891, 6249, 6044, 3176, 733, 1, 5, 65, 550, 3000, 10061, 21524, 29133, 24302, 11326, 2282
EXAMPLE
Triangle begins: (n >= 3, k >= 1)
1;
1, 1;
1, 1, 1;
1, 2, 3, 3;
1, 2, 6, 7, 4;
1, 3, 11, 24, 24, 12;
1, 3, 17, 51, 89, 74, 27;
1, 4, 26, 109, 265, 371, 259, 82;
1, 4, 36, 194, 660, 1291, 1478, 891, 228;
...
PROG
(PARI) \\ See A295419 for DissectionsModDihedral() T=DissectionsModDihedral(apply(i->y, [1..12]));
for(n=3, #T, for(k=1, n-2, print1(polcoeff(T[n], k), ", ")); print)
Number of nonequivalent dissections of an n-gon into n-4 polygons by nonintersecting diagonals up to rotation.
(Formerly M1859)
+10
6
1, 2, 8, 40, 165, 712, 2912, 11976, 48450, 195580, 784504, 3139396, 12526605, 49902440, 198499200, 788795924, 3131945190, 12428258796, 49295766000, 195464345440, 774857314042, 3071175790232, 12171403236288, 48233597481200, 191138095393700, 757436171945952
COMMENTS
In other words, the number of (n-5)-dissections of an n-gon modulo the cyclic action.
Equivalently, the number of two-dimensional faces of the (n-3)-dimensional associahedron modulo the cyclic action.
The dissection will always be composed of either 1 pentagon and n-5 triangles or 2 quadrilaterals and n-6 triangles. - Andrew Howroyd, Nov 24 2017
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
MAPLE
C:=n->binomial(2*n, n)/(n+1);
T31:=proc(n) local t1; global C;
t1 := (n-3)^2*(n-4)*C(n-2)/(4*n*(2*n-5));
if n mod 5 = 0 then t1:=t1+(4/5)*C(n/5-1) fi;
if n mod 2 = 0 then t1:=t1+(n-4)*C(n/2-1)/8 fi;
t1; end;
[seq(T31(n), n=5..40)];
MATHEMATICA
Table[t1 = (n - 3)^2*(n - 4)*CatalanNumber[n - 2]/(4*n*(2*n - 5)); If[Mod[n, 5] == 0, t1 = t1 + (4/5)*CatalanNumber[n/5 - 1]]; If[Mod[n, 2] == 0, t1 = t1 + (n - 4)*CatalanNumber[n/2 - 1]/8]; t1, {n, 5, 20}] (* T. D. Noe, Jan 03 2013 *)
PROG
(PARI) \\ See A295495 for DissectionsModCyclic() { my(v=DissectionsModCyclic(apply(i->if(i>=3&&i<=5, y^(i-3) + O(y^3)), [1..30]))); apply(p->polcoeff(p, 2), v[5..#v]) } \\ Andrew Howroyd, Nov 24 2017
Number of nonequivalent dissections of an n-gon into n-3 polygons by nonintersecting diagonals up to rotation.
+10
6
1, 1, 4, 12, 43, 143, 504, 1768, 6310, 22610, 81752, 297160, 1086601, 3991995, 14732720, 54587280, 202997670, 757398510, 2834510744, 10637507400, 40023636310, 150946230006, 570534578704, 2160865067312, 8199711378716, 31170212479588, 118686578956272
COMMENTS
This is almost identical to A003444, but has a different offset and a more precise definition. In other words, the number of almost-triangulations of an n-gon modulo the cyclic action.
Equivalently, the number of edges of the (n-3)-dimensional associahedron modulo the cyclic action.
The dissection will always be composed of one quadrilateral and n-4 triangles. - Andrew Howroyd, Nov 25 2017 Also number of necklaces of 2 colors with 2n-4 beads and n black ones. - Wouter Meeussen, Aug 03 2002
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(n) = (1/(2n-4)) Sum_{d |(2n-4, n)} phi(d)*binomial((2n-4)/d, n/d) for n >= 4. - Wouter Meeussen, Aug 03 2002
MAPLE
C:=n->binomial(2*n, n)/(n+1);
T2:= proc(n) local t1; global C;
t1 := (n-3)*C(n-2)/(2*n);
if n mod 4 = 0 then t1:=t1+C(n/4-1)/2 fi;
if n mod 2 = 0 then t1:=t1+C(n/2-1)/4 fi;
t1; end;
[seq(T2(n), n=4..40)];
MATHEMATICA
c[n_] := Binomial[2*n, n]/(n+1);
T2[n_] := Module[{t1}, t1 = (n-3)*c[n-2]/(2*n); If[Mod[n, 4] == 0, t1 = t1 + c[n/4-1]/2]; If[Mod[n, 2] == 0, t1 = t1 + c[n/2-1]/4]; t1];
a[n_] := Sum[EulerPhi[d]*Binomial[(2n-4)/d, n/d], {d, Divisors[GCD[2n-4, n] ]}]/(2n-4);
PROG
(PARI)
a(n) = if(n>=4, sumdiv(gcd(2*n-4, n), d, eulerphi(d)*binomial((2*n-4)/d, n/d))/(2*n-4)) \\ Andrew Howroyd, Nov 25 2017
Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals up to rotation.
(Formerly M3330)
+10
4
1, 4, 8, 16, 25, 40, 56, 80, 105, 140, 176, 224, 273, 336, 400, 480, 561, 660, 760, 880, 1001, 1144, 1288, 1456, 1625, 1820, 2016, 2240, 2465, 2720, 2976, 3264, 3553, 3876, 4200, 4560, 4921, 5320, 5720, 6160, 6601, 7084, 7568, 8096, 8625, 9200, 9776, 10400
COMMENTS
In other words, the number of 2-dissections of an n-gon modulo the cyclic action.
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
G.f.: x^5 * (1 + 2*x - x^2 ) / ((1 - x)^4*(1 + x)^2).
See also the Maple code for an explicit formula.
a(n) = (n-4)*(2*n^2-4*n-3*(1-(-1)^n))/24, for n>=5. - Luce ETIENNE, Apr 04 2015
MAPLE
T51:= proc(n)
if n mod 2 = 0 then n*(n-2)*(n-4)/12;
else (n+1)*(n-3)*(n-4)/12; fi end;
MATHEMATICA
Table[((n - 4) (2 n^2 - 4 n - 3 (1 - (-1)^n)) / 24), {n, 5, 60}] (* Vincenzo Librandi, Apr 05 2015 *) CoefficientList[Series[(1+2*x-x^2)/((1-x)^4*(1+x)^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 05 2015 *)
PROG
(PARI) Vec((1 + 2*x - x^2 ) / ((1 - x)^4*(1 + x)^2) + O(x^50)) \\ Michel Marcus, Apr 04 2015 (PARI) \\ See A295495 for DissectionsModCyclic() { my(v=DissectionsModCyclic(apply(i->y, [1..30]))); apply(p->polcoeff(p, 3), v[5..#v]) } \\ Andrew Howroyd, Nov 24 2017 (Magma) [(n-4)*(2*n^2-4*n-3*(1-(-1)^n))/24: n in [5..60]]; // Vincenzo Librandi, Apr 05 2015
Number of embeddings on the sphere of Halin graphs on n unlabeled nodes up to orientation-preserving homeomorphisms.
+10
3
0, 0, 0, 1, 1, 2, 2, 4, 7, 16, 32, 76, 181, 443, 1098, 2793, 7127, 18458, 48128, 126580, 334955, 892187, 2388674, 6428489, 17377599, 47174939, 128555088, 351580903, 964696719, 2655197386, 7329051870, 20284610084, 56283140111, 156537249660, 436338547904, 1218824493990, 3411297202411
COMMENTS
Halin graphs are planar and 3-connected and can be embedding in the sphere in essentially one way up to mirror symmetry. This sequence counts each graph as either 1 or 2 depending on if it is mirror symmetric.
PROG
(PARI) A380360seq(36) \\ See PARI Link in A380362 for program code.
Triangle read by rows: T(n,k) is the number of embeddings on the sphere of Halin graphs on n unlabeled nodes with circuit rank k up to orientation-preserving homeomorphisms, 3 <= k <= n-1.
+10
3
1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 0, 4, 2, 1, 0, 0, 0, 4, 8, 3, 1, 0, 0, 0, 0, 12, 16, 3, 1, 0, 0, 0, 0, 6, 40, 25, 4, 1, 0, 0, 0, 0, 0, 43, 93, 40, 4, 1, 0, 0, 0, 0, 0, 19, 165, 197, 56, 5, 1, 0, 0, 0, 0, 0, 0, 143, 505, 364, 80, 5, 1
COMMENTS
The circuit rank is equal to the number of leaves on the tree before it is extended into a Halin graph by joining up the leaves.
The main diagonal of the graph corresponds with the wheel graphs which have the greatest circuit rank of all Halin graphs.
T(n,k) is also the number of nonequivalent dissections of a k-gon into n-k polygons by nonintersecting diagonals up to rotation.
EXAMPLE
Triangle begins:
n\k| 3 4 5 6 7 8 9 10 11 12 13
-----+-----------------------------------------
4 | 1;
5 | 0, 1;
6 | 0, 1, 1;
7 | 0, 0, 1, 1;
8 | 0, 0, 1, 2, 1;
9 | 0, 0, 0, 4, 2, 1;
10 | 0, 0, 0, 4, 8, 3, 1;
11 | 0, 0, 0, 0, 12, 16, 3, 1;
12 | 0, 0, 0, 0, 6, 40, 25, 4, 1;
13 | 0, 0, 0, 0, 0, 43, 93, 40, 4, 1;
14 | 0, 0, 0, 0, 0, 19, 165, 197, 56, 5, 1;
...
PROG
(PARI) \\ See PARI Link in A380362 for program code. { my(T=A380361rows(12)); for(i=1, #T, print(T[i])) }
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