A215341
Expansion of series_reversion( x/(1+x^4*sum(k>=0, x^k)) ) / x.
Original entry on oeis.org
1, 0, 0, 0, 1, 1, 1, 1, 5, 10, 16, 23, 53, 118, 232, 411, 813, 1718, 3568, 7012, 13925, 28603, 59533, 121878, 247915, 509136, 1057278, 2194138, 4536943, 9394145, 19552639, 40803472, 85131237, 177640486, 371426592, 778275264, 1632420197, 3425607187, 7195476245, 15134138683, 31866093569
Offset: 0
Cf.
A000108 (rev. of x/(1+1*sum(k>=1,x^k)) ),
A005043 (rev. of x/(1+x*sum(k>=1,x^k)) ),
A114997 (rev. of x/(1+x^2*sum(k>=1,x^k)) ).
Cf.
A001003 (rev. of x*(1-1*sum(k=1,N,x^k)) ),
A046736 (rev. of x*(1-x*sum(k=1,N,x^k)) ),
A054514 (rev. of x*(1-x^2*sum(k=1,N,x^k)) ),
A215342 (rev. of x*(1-x^3*sum(k=1,N,x^k)) ).
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b:= proc(x, y, t) option remember; `if`(y0 and t in [0, 4],
b(x-1, y, 0), 0) +b(x, y-1, min(t+1, 4))))
end:
a:= n-> b(n, n, 0):
seq(a(n), n=0..50); # Alois P. Heinz, Apr 16 2013
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InverseSeries[x/(1+x^4/(1-x)) + O[x]^50] // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Mar 29 2017 *)
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a(n):=sum(binomial(n+1,i)*binomial(n-3*i-1,n-4*i),i,0,floor(n/4))/(n+1); /* Vladimir Kruchinin, Apr 01 2019 */
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N=66; Vec( serreverse(x/(1+x^4*sum(k=0,N,x^k))+O(x^N)) / x )
Modified definition to obtain offset 0 for combinatorial interpretation,
Joerg Arndt, Apr 16 2013
A215342
Expansion of series reversion of x*(1-x^3*sum(k>=1, x^k)).
Original entry on oeis.org
1, 0, 0, 0, 1, 1, 1, 1, 6, 12, 19, 27, 71, 166, 329, 579, 1222, 2756, 5921, 11754, 24179, 52372, 114031, 239726, 502269, 1074961, 2333143, 5017552, 10714567, 23006558, 49861081, 108122488, 233691980, 505329915, 1097463037, 2389325284, 5199960642, 11314793335, 24663217250, 53864633059
Offset: 1
Use the Lang and the Abramowitz and Stegun links in A111785. In the A-S list of partitions of the integer n on page 831 null all partitions containing 1, 2, or 3. These correspond to the null coefficients of x^2, x^3, and x^4 in the series to be reverted and to 3-, 4-, and 5-gons not being allowed in the dissections. a(9)=6 corresponds to the A-S partitions (n=8,m=1, partition 1)=8 and (8,2,4)=4^2, and these in turn correspond to one undissected 10-gon + five ways to divide a 10-gon into two 6-gons. a(10)=12 corresponds to (9,1,1)=9 and (9,2,4)=4,5, corresponding to one undissected 11-gon + the eleven ways to divide an 11-gon into a 6-gon and 7-gon. - _Tom Copeland_, Feb 15 2014
- Vaclav Kotesovec, Table of n, a(n) for n = 1..250
- Alison Schuetz, Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO]
- D. Birmajer, J. B. Gil, M. D. Weiner, Colored partitions of a convex polygon by noncrossing diagonals, arXiv preprint arXiv:1503.05242 [math.CO], 2015.
Cf.
A001003 (rev. of x*(1-1*sum(k>=1,x^k)) ),
A046736 (rev. of x*(1-x*sum(k>=1,x^k)) ),
A054514 (rev. of x*(1-x^2*sum(k>=1,x^k)) ).
Cf.
A000108 (rev. of x/(1+1*sum(k>=1,x^k)) ),
A005043 (rev. of x/(1+x*sum(k>=1,x^k)) ),
A114997 (rev. of x/(1+x^2*sum(k>=1,x^k)) ),
A215341 (rev. of x/(1+x^3*sum(k>=1,x^k)) ).
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nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=0; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[1-(1+x)*AGF+x*AGF^2 +x^4*AGF^5,x,j]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Mar 23 2014 *)
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N=66; Vec(serreverse(x*(1-x^3*sum(k=1,N,x^k))+O(x^N)))
A350116
Number of ways to partition the set of vertices of a convex {n+8}-gon into 3 non-intersecting polygons.
Original entry on oeis.org
0, 12, 45, 110, 220, 390, 637, 980, 1440, 2040, 2805, 3762, 4940, 6370, 8085, 10120, 12512, 15300, 18525, 22230, 26460, 31262, 36685, 42780, 49600, 57200, 65637, 74970, 85260, 96570, 108965, 122512, 137280, 153340, 170765, 189630, 210012, 231990, 255645, 281060, 308320
Offset: 0
The a(1) = 12 solutions are:
{123}{456}{789}, {234}{567}{891}, {345}{678}{912},
{156}{234}{567}, {267}{345}{891}, {378}{456}{912},
{489}{567}{123}, {591}{678}{234}, {612}{789}{345},
{723}{891}{456}, {834}{912}{567}, {945}{123}{678}.
In the above, the numbers can be considered to be the partition of a 9-set into 3 blocks or the partition of the vertices of a convex 9-gon into 3 triangles (with the vertices labeled 1..9 in order).
a(2) = 45 corresponding to the number of ways to partition the vertices of a 10-gon into two triangles and one quadrilateral.
The case of any number of parts for an n-gon is
A114997.
The case of exactly 2 parts for a {n+5}-gon is
A055998.
A350248
Triangle read by rows: T(n,k) is the number of noncrossing partitions of an n-set into k blocks of size 3 or more, n >= 0, 0 <= k <= floor(n/3).
Original entry on oeis.org
1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 3, 0, 1, 7, 0, 1, 12, 0, 1, 18, 12, 0, 1, 25, 45, 0, 1, 33, 110, 0, 1, 42, 220, 55, 0, 1, 52, 390, 286, 0, 1, 63, 637, 910, 0, 1, 75, 980, 2275, 273, 0, 1, 88, 1440, 4900, 1820, 0, 1, 102, 2040, 9520, 7140, 0, 1, 117, 2805, 17136, 21420, 1428
Offset: 0
Triangle begins:
1;
0;
0;
0, 1;
0, 1;
0, 1;
0, 1, 3;
0, 1, 7;
0, 1, 12;
0, 1, 18, 12;
0, 1, 25, 45;
0, 1, 33, 110;
0, 1, 42, 220, 55;
0, 1, 52, 390, 286;
0, 1, 63, 637, 910;
0, 1, 75, 980, 2275, 273;
0, 1, 88, 1440, 4900, 1820;
0, 1, 102, 2040, 9520, 7140;
...
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T(n)={my(p=1+O(x^3)); for(i=1, n\3, p=1+y*(x*p)^3/(1-x*p)); [Vecrev(t)| t<-Vec(p + O(x*x^n))]}
{my(A=T(12)); for(i=1, #A, print(A[i]))}
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T(n,k) = if(n==0 || k>n\3, k==0, binomial(n+1, n-k+1) * binomial(n-2*k-1, k-1) / (n+1)) \\ Andrew Howroyd, Dec 31 2021
A366052
Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1-x+x^3) ).
Original entry on oeis.org
1, 2, 7, 31, 154, 819, 4559, 26226, 154664, 930040, 5680920, 35150493, 219850505, 1387717660, 8828668582, 56553846890, 364449091112, 2361118198094, 15369247139879, 100468188756849, 659271433474584, 4341140182940382, 28675590236716905
Offset: 0
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a(n) = sum(k=0, n\3, binomial(n+1, k)*binomial(3*n-2*k+1, n-3*k))/(n+1);
A365702
G.f. satisfies A(x) = 1 + x^5*A(x)^5 / (1 - x*A(x)).
Original entry on oeis.org
1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 6, 12, 19, 27, 36, 81, 177, 341, 592, 951, 1726, 3417, 6766, 12812, 22951, 41531, 78222, 151291, 291957, 550015, 1024683, 1924543, 3671017, 7063893, 13532120, 25730347, 48840523, 93154161, 178806493, 343926597, 660308308, 1265195467
Offset: 0
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a(n) = sum(k=0, n\5, binomial(n-4*k-1, n-5*k)*binomial(n+1, k))/(n+1);
A365724
G.f. satisfies A(x) = 1 + x^3*A(x)^3*(1 + x*A(x)).
Original entry on oeis.org
1, 0, 0, 1, 1, 0, 3, 7, 4, 12, 45, 55, 77, 286, 546, 728, 1960, 4760, 7548, 15504, 39729, 75582, 140448, 336490, 723327, 1366200, 2992990, 6758895, 13522275, 28094040, 63183315, 133231800, 273896532, 600805296, 1305229332, 2720740792, 5843241088, 12797739672
Offset: 0
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a(n) = sum(k=0, n\3, binomial(k, n-3*k)*binomial(n+1, k))/(n+1);
A366051
Expansion of (1/x) * Series_Reversion( x/(1-x+x^3) ).
Original entry on oeis.org
1, -1, 1, 0, -3, 9, -16, 13, 29, -157, 391, -562, -32, 3002, -10373, 20747, -18083, -47941, 271117, -712216, 1066699, 122131, -6464446, 22907125, -46951992, 40883304, 120187926, -679375906, 1809757015, -2731745887, -468147579, 17768126376, -63256877763
Offset: 0
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a(n) = sum(k=0, n\3, (-1)^(n-k)*binomial(n+1, k)*binomial(n-k+1, n-3*k))/(n+1);
A365695
G.f. satisfies A(x) = 1 + x^3*A(x)^5 / (1 - x*A(x)).
Original entry on oeis.org
1, 0, 0, 1, 1, 1, 6, 12, 19, 62, 156, 318, 852, 2254, 5262, 13441, 35543, 88772, 226880, 596937, 1539188, 3980364, 10468270, 27410289, 71702956, 189169352, 499529048, 1318355542, 3493861461, 9278408639, 24647900618, 65620808508, 175037591303, 467277998136
Offset: 0
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a(n) = sum(k=0, n\3, binomial(n-2*k-1, n-3*k)*binomial(n+2*k+1, k)/(n+2*k+1));
A366053
Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1-x+x^3) ).
Original entry on oeis.org
1, 3, 15, 92, 628, 4579, 34917, 275041, 2220472, 18275896, 152780718, 1293657534, 11072033677, 95629771059, 832460471465, 7296161486583, 64331378963164, 570228657335744, 5078345448484216, 45418278349485960, 407749837317844851, 3673300856466182388
Offset: 0
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a(n) = sum(k=0, n\3, binomial(n+1, k)*binomial(4*n-2*k+2, n-3*k))/(n+1);
Showing 1-10 of 13 results.
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