A054514 -id:A054514 - cp's OEIS frontend

A215341 Expansion of series_reversion( x/(1+x^4*sum(k>=0, x^k)) ) / x.

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

Joerg Arndt, Aug 19 2012

nonn

Number of Dyck words of semilength n with substrings UUU...UU (ascents) only of lengths >= 4. See A215340 for an explanation. [Joerg Arndt, Apr 16 2013]

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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)) ).

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

InverseSeries[x/(1+x^4/(1-x)) + O[x]^50] // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Mar 29 2017 *)

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 */

N=66; Vec( serreverse(x/(1+x^4*sum(k=0,N,x^k))+O(x^N)) / x )

G.f. A(x) satisfies 0 = -x^4*A(x)^4 - x*A(x)^2 + (x + 1)*A(x) - 1. [Joerg Arndt, Mar 01 2014]
Recurrence: 2*n*(n+1)*(2*n+3)*(16204*n^4 - 82948*n^3 + 139973*n^2 - 85643*n + 10674)*a(n) =  - (n-1)*n*(307876*n^5 - 960260*n^4 + 288863*n^3 + 582749*n^2 + 5406*n + 12696)*a(n-1) + 4*(n-2)*(129632*n^6 - 469136*n^5 + 354226*n^4 + 317255*n^3 - 469674*n^2 + 176517*n - 21420)*a(n-2) - 2*(n-3)*(n-2)*(16204*n^5 - 34336*n^4 + 82943*n^3 - 208775*n^2 + 192120*n - 40656)*a(n-3) + 6*(n-3)*(n-2)*(97224*n^5 - 351852*n^4 + 179198*n^3 + 540009*n^2 - 571727*n + 92968)*a(n-4) + 229*(n-4)*(n-3)*(n-2)*(16204*n^4 - 18132*n^3 - 11647*n^2 + 10275*n - 1740)*a(n-5). - Vaclav Kotesovec, Mar 22 2014
a(n) ~ sqrt((s-1)*s^3/(6-8*s+3*s^2)) / (2*sqrt(Pi)*n^(3/2)*r^n), where r = 0.4577644245749322..., s = 1.232809919151165... are roots of the system of equations 1 + r*s^2 + r^4*s^4 = (1+r)*s, 1+r = 2*r*s + 4*r^4*s^3. - Vaclav Kotesovec, Mar 22 2014
a(n) = (1/(n+1)) * Sum_{i=0..floor(n/4)} C(n+1,i) * C(n-3*i-1,n-4*i). - Vladimir Kruchinin, Apr 01 2019

Modified definition to obtain offset 0 for combinatorial interpretation, Joerg Arndt, Apr 16 2013

A054515 Number of ways to place non-intersecting diagonals in convex (n+2)-gon so as to create no quadrilaterals.

Original entry on oeis.org

1, 1, 2, 6, 21, 78, 301, 1198, 4888, 20340, 85986, 368239, 1594183, 6965380, 30675399, 136026759, 606848034, 2721783023, 12265670909, 55511013680, 252193872912, 1149742659556, 5258257323304, 24117924005616, 110915268468358, 511334146237807, 2362650323603539

Offset: 0

Len Smiley, Apr 08 2000

nonn

Number of tree interval posets of permutations with n+1 minimal elements. - Mathilde Bouvel, Oct 21 2021

			a(3) = 6 because the pentagon allows null placement and five ways to place two diagonals.

Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, Blockwise simple permutations, arXiv:2303.13115 [math.CO], 2023.
Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, Geometric view of interval poset permutations, arXiv:2411.13193 [math.CO], 2024. See pp. 3, 8.
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, Colored partitions of a convex polygon by noncrossing diagonals, arXiv preprint arXiv:1503.05242 [math.CO], 2015
Mathilde Bouvel, Lapo Cioni, and Benjamin Izart, The interval posets of permutations seen from the decomposition tree perspective, arXiv:2110.10000 [math.CO], 2021.
Len Smiley, Generalization and some variants, see Quad-free.
Bridget Eileen Tenner, Interval posets for permutations, arXiv:2007.06142 [math.CO], 2020-2021.

Cf. A046736, A049124, A003168, A054514, A348479 (free interv. posets not necess. trees).

read("transforms") :
taylor( (1-2*y+y^2-y^3)/(1-y),y=0,50) ;
gfun[seriestolist](%) ;
REVERT(%) ; # R. J. Mathar, Nov 04 2021

InverseSeries[Series[(y-2*y^2+y^3-y^4)/(1-y), {y, 0, 24}], x] (* then A(x)=[y(x)-x]/x *)

my(N=28, x='x+O('x^N)); Vec(serreverse((x-2*x^2+x^3-x^4)/(1-x))) \\ Hugo Pfoertner, Jan 26 2024

REVERT transform of (1-2*x+x^2-x^3)/(1-x) [Smiley].
a(n-1) = (1/n) * [binomial(2n-2,n-1) + Sum_{i=1..(n-3)} Sum_{k=1..Min(i,(n-i-1)/2)} binomial(n+i-1,i)*binomial(i,k)*binomial(n-i-k-2,k-1) ] if n>1. Proved in M. Bouvel, L. Cioni, B. Izart (Theorem 21) with offset 1. - Mathilde Bouvel, Oct 21 2021
G.f. A(z) = Sum_{n>=0} a(n)*z^n satisfies A(z) = 1 + z*A^2 + z^3*A^4/(1-z*A). Proved in M. Bouvel, L. Cioni, B. Izart (Equation (6) page 17 with offset 1). - Mathilde Bouvel, Oct 21 2021
Asymptotic behavior of a(n-1) is c*n^(-3/2)*r^n with c approximately 0.0792 and r approximately 4.8920. Proved in M. Bouvel, L. Cioni, B. Izart (Theorem 22). - Mathilde Bouvel, Oct 21 2021
D-finite with recurrence 23 *n *(n-1) *(12869043*n-33144451) *(n+1) *a(n) -n *(n-1) *(1989552043*n^2-6117767430*n+2643232213) * a(n-1) +(n-1) *(3359030609*n^3-15361701516*n^2+20123332181*n-6949961920) *a(n-2) +(-3560897749*n^4+25182507306*n^3-62054513365*n^2 +60006265908*n-16495478980) *a(n-3) +3*(146027817*n^4-1247820696*n^3+3378236999*n^2-2363753280*n-1468123920)*a(n-4) -3*(335627*n+695280) *(3*n-13) *(3*n-11) *(n-4) *a(n-5)=0. - R. J. Mathar, Oct 28 2021
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(n+k,k) * binomial(2*n-k,n-3*k). - Seiichi Manyama, Jan 26 2024

a(0) = 1 prefixed by R. J. Mathar, Nov 04 2021

A215342 Expansion of series reversion of x(1-x^3sum(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

Joerg Arndt, Aug 19 2012

nonn

Dissections (using non-intersecting diagonals) of a convex (n+1)-gon into k-gons where k>=6. [Joerg Arndt, Feb 15 2014]

			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)) ).

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 *)

N=66; Vec(serreverse(x*(1-x^3*sum(k=1,N,x^k))+O(x^N)))

Recurrence: 283*(n-3)*(n-2)*(n-1)*n*(14438110231*n^6 - 346214993274*n^5 + 3438949212625*n^4 - 18105364836570*n^3 + 53265099505324*n^2 - 82987438028496*n + 53465930027280)*a(n) = (n-3)*(n-2)*(n-1)*(5399853226394*n^7 - 137584187324067*n^6 + 1483504918415939*n^5 - 8763694066910355*n^4 + 30585682233578711*n^3 - 62946796681030518*n^2 + 70573456271906136*n - 33158656683118080)*a(n-1) - (n-3)*(n-2)*(534210078547*n^8 - 14946795065326*n^7 + 180235890644998*n^6 - 1221993860476624*n^5 + 5087726442447403*n^4 - 13295664719568394*n^3 + 21246368278875372*n^2 - 18919520411340456*n + 7154560952974080)*a(n-2) - 5*(n-4)*(n-3)*(28876220462*n^8 - 793496758165*n^7 + 9354947999333*n^6 - 61627542806839*n^5 + 247116200695877*n^4 - 613894185501244*n^3 + 913857055091496*n^2 - 732955177968120*n + 234541607788800)*a(n-3) + 5*(9947857949159*n^10 - 357916425755694*n^9 + 5753280746412201*n^8 - 54388490463504720*n^7 + 334719732595671573*n^6 - 1400557913088383070*n^5 + 4032929135663406319*n^4 - 7886242788829977540*n^3 + 10015113186875731788*n^2 - 7452248915385205056*n + 2464721951024954880)*a(n-4) - 8*(n-5)*(2*n - 9)*(4*n - 21)*(4*n - 19)*(14438110231*n^6 - 259586331888*n^5 + 1924445899720*n^4 - 7522955714190*n^3 + 16337121992089*n^2 - 18661982982042*n + 8745398997120)*a(n-5). - Vaclav Kotesovec, Mar 22 2014
a(n) ~ s*sqrt((3*r*s+r-4)/(5*(3*r*s-2*r-2))) / (2*sqrt(Pi) * n^(3/2) * r^(n-1)), where s = 1.1954869989505368389... is the root of the equation 64 - 897*s + 2460*s^2 - 2787*s^3 + 1442*s^4 - 283*s^5 = 0, and r = (4*s-5)/(s*(3*s-4)) = 0.441061092405258554919... - Vaclav Kotesovec, Mar 23 2014
G.f. A(x) for offset 0 satisfies 1-(1+x)*A(x) + x*A(x)^2 + x^4*A(x)^5 = 0. - Vaclav Kotesovec, Mar 23 2014

A369011 Expansion of (1/x) * Series_Reversion( x * (1-x^3/(1-x))^2 ).

Original entry on oeis.org

1, 0, 0, 2, 2, 2, 17, 36, 59, 240, 669, 1452, 4538, 13574, 34505, 99816, 299112, 825768, 2364715, 7023466, 20182611, 58327250, 172491553, 505163444, 1476966513, 4370772096, 12924382671, 38149522136, 113266357609, 336894290910, 1001473479313, 2985508193930

Offset: 0

Seiichi Manyama, Jan 11 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for reversions of series

Cf. A211789, A368957.
Cf. A054514, A369014.
Cf. A368962, A368966, A368968.

my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1-x^3/(1-x))^2)/x)

a(n, s=3, t=2, u=-2) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(2*n+k+1,k) * binomial(n-2*k-1,n-3*k).

A366071 Expansion of (1/x) * Series_Reversion( x*(1+x-x^3)/(1+x) ).

Original entry on oeis.org

1, 0, 0, 1, -1, 1, 3, -8, 14, 1, -49, 144, -162, -139, 1159, -2532, 2036, 6062, -26282, 47440, -11474, -190071, 606163, -838984, -481092, 5479390, -13618658, 13030368, 28786262, -148598623, 294393355, -128639411, -1086088045, 3848604261, -5935686369, -1750697623

Offset: 0

Seiichi Manyama, Sep 29 2023

sign

Cf. A366095, A366096, A366097.
Cf. A366101, A366102.
Cf. A054514.

a(n) = sum(k=0, n\3, (-1)^(n-k)*binomial(n+k, k)*binomial(n-2*k-1, n-3*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (-1)^(n-k) * binomial(n+k,k) * binomial(n-2*k-1,n-3*k).

A369014 Expansion of (1/x) * Series_Reversion( x * (1-x^3/(1-x))^3 ).

Original entry on oeis.org

1, 0, 0, 3, 3, 3, 36, 78, 129, 685, 2043, 4554, 17233, 57279, 153045, 509848, 1724739, 5117643, 16445555, 55165536, 173225715, 555899673, 1847495415, 5971507824, 19333284247, 63975307425, 209807070669, 685973054145, 2269660792842, 7501194321663, 24725092907853

Offset: 0

Seiichi Manyama, Jan 11 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for reversions of series

Cf. A369012, A369013.

Cf. A054514, A369011.

my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1-x^3/(1-x))^3)/x)

a(n, s=3, t=3, u=-3) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(3*n+k+2,k) * binomial(n-2*k-1,n-3*k).

A365725 G.f. satisfies A(x) = 1 + x^3A(x)^4(1 + x*A(x)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 4, 9, 5, 22, 78, 91, 175, 680, 1224, 1938, 6270, 14630, 24794, 63756, 166980, 322920, 720720, 1900080, 4125888, 8803008, 22151360, 51778804, 111882100, 267682272, 645736432, 1442390092, 3346519020, 8094247798, 18657762006, 42890295734

Offset: 0

Seiichi Manyama, Sep 17 2023

nonn

Index entries for reversions of series

Cf. A308616, A365723, A365724, A365726.
Cf. A001002, A217358, A365731.
Cf. A054514.

a(n) = sum(k=0, n\3, binomial(k, n-3*k)*binomial(n+k+1, k)/(n+k+1));

a(n) = Sum_{k=0..floor(n/3)} binomial(k,n-3*k) * binomial(n+k+1,k) / (n+k+1).

G.f.: (1/x) * Series_Reversion( x*(1 - x^3*(1 + x)) ). - Seiichi Manyama, Sep 24 2024

A368931 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x-x^3) ).

Original entry on oeis.org

1, 2, 7, 31, 154, 819, 4560, 26244, 154874, 932074, 5698745, 35297535, 221016593, 1396717756, 8896798020, 57062237502, 368201804973, 2388587515239, 15568995139404, 101913055166811, 669678357109300, 4415837460391845, 29210203356645090

Offset: 0

Seiichi Manyama, Jan 10 2024

nonn

Cf. A236339, A368932, A368933.

Cf. A049140, A054514.

a(n) = sum(k=0, n\3, binomial(n+k, k)*binomial(3*n-2*k+1, n-3*k))/(n+1);

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)*(1-x-x^3))/x)

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(n+k,k) * binomial(3*n-2*k+1,n-3*k).

A365695 G.f. satisfies A(x) = 1 + x^3A(x)^5 / (1 - xA(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

Seiichi Manyama, Sep 16 2023

nonn

Cf. A023426, A023432, A054514, A114997, A365694.

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));

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k-1,n-3*k) * binomial(n+2*k+1,k) / (n+2*k+1).

A365694 G.f. satisfies A(x) = 1 + x^3A(x)^2 / (1 - xA(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

Seiichi Manyama, Sep 16 2023

nonn

Cf. A023426, A023432, A054514, A114997, A365695.

Cf. A365243.

CoefficientList[Series[2/(1 + x + Sqrt[1 + x*(-2 + x - 4*x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 16 2023 *)

a(n) = sum(k=0, n\3, binomial(n-2*k-1, n-3*k)*binomial(n-k+1, k)/(n-k+1));

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k-1,n-3*k) * binomial(n-k+1,k) / (n-k+1).

G.f.: A(x) = 2/(1 + x + sqrt(1 + x*(-2 + x - 4*x^2))). - Vaclav Kotesovec, Sep 16 2023

cp's OEIS Frontend

A215341 Expansion of series_reversion( x/(1+x^4*sum(k>=0, x^k)) ) / x.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

Extensions

A054515 Number of ways to place non-intersecting diagonals in convex (n+2)-gon so as to create no quadrilaterals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A215342 Expansion of series reversion of x*(1-x^3*sum(k>=1, x^k)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A369011 Expansion of (1/x) * Series_Reversion( x * (1-x^3/(1-x))^2 ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A366071 Expansion of (1/x) * Series_Reversion( x*(1+x-x^3)/(1+x) ).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A369014 Expansion of (1/x) * Series_Reversion( x * (1-x^3/(1-x))^3 ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A365725 G.f. satisfies A(x) = 1 + x^3*A(x)^4*(1 + x*A(x)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A368931 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x-x^3) ).

Views

A215342 Expansion of series reversion of x(1-x^3sum(k>=1, x^k)).

A365725 G.f. satisfies A(x) = 1 + x^3A(x)^4(1 + x*A(x)).

A365695 G.f. satisfies A(x) = 1 + x^3A(x)^5 / (1 - xA(x)).

A365694 G.f. satisfies A(x) = 1 + x^3A(x)^2 / (1 - xA(x)).