A049124 -id:A049124 - cp's OEIS frontend

A003168 Number of blobs with 2n+1 edges.

1, 1, 4, 21, 126, 818, 5594, 39693, 289510, 2157150, 16348960, 125642146, 976789620, 7668465964, 60708178054, 484093913917, 3884724864390, 31348290348086, 254225828706248, 2070856216759478, 16936016649259364

Offset: 0

N. J. A. Sloane

a(n) is the number of ways to dissect a convex (2n+2)-gon with non-crossing diagonals so that no (2m+1)-gons (m>0) appear. - Len Smiley
a(n) is the number of plane trees with 2n+1 leaves and all non-leaves having an odd number > 1 of children. - Jordan Tirrell, Jun 09 2017
a(n) is the number of noncrossing cacti with n+1 nodes. See A361242. - Andrew Howroyd, Mar 07 2023

			a(2)=4 because we may place exactly one diagonal in 3 ways (forming 2 quadrilaterals), or not place any (leaving 1 hexagon).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..100 from T. D. Noe)
Thomas H. Bertschinger, Joseph Slote, Olivia Claire Spencer, and Samuel Vinitsky, The Mathematics of Origami, Undergrad Thesis, Carleton College (2016).
D. Birmajer, J. B. Gil, and M. D. Weiner, Colored partitions of a convex polygon by noncrossing diagonals, arXiv preprint arXiv:1503.05242 [math.CO], 2015.
L. Carlitz, Enumeration of two-line arrays, Fib. Quart., Vol. 11 Number 2 (1973), 113-130.
Frédéric Chapoton and Philippe Nadeau, Combinatorics of the categories of noncrossing partitions, Séminaire Lotharingien de Combinatoire 78B (2017), Article #37.
Michael Drmota, Anna de Mier, and Marc Noy, Extremal statistics on non-crossing configurations, Discrete Math. 327 (2014), 103--117. MR3192420. See p. 116, B_b(z). - N. J. A. Sloane, May 18 2014
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 415
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Jean-Christophe Novelli and Jean-Yves Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
Jean-Christophe Novelli and Jean-Yves Thibon, Noncommutative Symmetric Functions and Lagrange Inversion II: Noncrossing partitions and the Farahat-Higman algebra, arXiv:2106.08257 [math.CO], 2021-2022.
Ronald C. Read, On the enumeration of a class of plane multigraphs, Aequat. Math. 31 (1986) No. 1, 47-63.
L. Smiley, Even-gon reference
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 30.

Cf. A049124 (no 2m-gons).
Cf. A003169, A100327, A114496, A007318, A361242.
Row sums of A102537, A243662. Column 2 of A336573.

import Data.List (transpose)
a003168 0 = 1
a003168 n = sum (zipWith (*)
   (tail $ a007318_tabl !! n)
   ((transpose $ take (3*n+1) a007318_tabl) !! (2*n+1)))
   `div` fromIntegral n
-- Reinhard Zumkeller, Oct 27 2013

Order := 40; solve(series((A-2*A^3)/(1-A^2),A)=x,A);
A003168 := n -> `if`(n=0,1,A100327(n)/2): seq(A003168(n),n=0..20); # Peter Luschny, Jun 10 2017

a[0] = 1; a[n_] = (2^(-n-1)*(3n)!* Hypergeometric2F1[-1-2n, -2n, -3n, -1])/((2n+1)* n!*(2n)!); Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 25 2011, after Vladimir Kruchinin *)

a(n)=if(n<0,0,polcoeff(serreverse((x-2*x^3)/(1-x^2)+O(x^(2*n+2))),2*n+1))

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=(1+x*A)/(1-x*A)^2); sum(k=0,n,polcoeff(A^(n-k),k))} \\ Paul D. Hanna, Nov 17 2004

seq(n) = Vec( 1 + serreverse(x/((1+2*x)*(1+x)^2) + O(x*x^n)) ) \\ Andrew Howroyd, Mar 07 2023

a(n) = Sum_{k=1..n} binomial(n, k)*binomial(2*n+k, k-1)/n.
G.f.: A(x) = Sum_{n>=0} a(n)*x^(2*n+1) satisfies (A-2*A^3)/(1-A^2)=x. - Len Smiley.
D-finite with recurrence 4*n*(2*n + 1)*(17*n - 22)*a(n) = (1207*n^3 - 2769*n^2 + 1850*n - 360)*a(n - 1) - 2*(17*n - 5)*(n - 2)*(2*n - 3)*a(n - 2). - Vladeta Jovovic, Jul 16 2004
G.f.: A(x) = 1/(1-G003169(x)) where G003169(x) is the g.f. of A003169. - Paul D. Hanna, Nov 17 2004
a(n) = JacobiP(n-1,1,n+1,3)/n for n > 0. - Mark van Hoeij, Jun 02 2010
a(n) = (1/(2*n+1))*Sum_{j=0..n} (-1)^j*2^(n-j)*binomial(2*n+1,j)*binomial(3*n-j,2*n). - Vladimir Kruchinin, Dec 24 2010
From Gary W. Adamson, Jul 08 2011: (Start)
a(n) = upper left term in M^n, M = the production matrix:
  1, 1
  3, 3, 1
  5, 5, 3, 1
  7, 7, 5, 3, 1
  9, 9, 7, 5, 3, 1
  ... (End)
a(n) ~ sqrt(14+66/sqrt(17)) * (71+17*sqrt(17))^n / (sqrt(Pi) * n^(3/2) * 2^(4*n+4)). - Vaclav Kotesovec, Jul 01 2015
From Peter Bala, Oct 05 2015: (Start)
a(n) = (1/n) * Sum_{i = 0..n} 2^(n-i-1)*binomial(2*n,i)* binomial(n,i+1).
O.g.f. = 1 + series reversion( x/((1 + 2*x)*(1 + x)^2) ).
Logarithmically differentiating the modified g.f. 1 + 4*x + 21*x^2 + 126*x^3 + 818*x^4 + ... gives the o.g.f. for A114496, apart from the initial term. (End)
G.f.: A(x) satisfies A = 1 + x*A^3/(1-x*A^2). - Jordan Tirrell, Jun 09 2017
a(n) = A100327(n)/2 for n>=1. - Peter Luschny, Jun 10 2017

A054514 Number of ways to place non-crossing diagonals in convex (n+4)-gon so as to create no triangles or quadrilaterals.

Original entry on oeis.org

1, 1, 1, 5, 10, 16, 45, 109, 222, 540, 1341, 3065, 7328, 18112, 43530, 105390, 260254, 639244, 1570257, 3893805, 9669236, 24014264, 59903650, 149806494, 374982790, 940835404, 2365679689, 5955973237, 15018854005, 37935575685, 95942896837, 242954350457, 616034170069, 1563810857705, 3974000543475

Offset: 1

Len Smiley, Apr 08 2000

nonn

			a(4)=5 because the octagon has the null placement and four ways to place a single diagonal.

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 p. 10.
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.
Len Smiley, Generalization and some variants

Cf. A046736, A049124, A003168, A054515.

f[x_] = InverseSeries[Series[(y - y^2 - y^4)/(1 - y), {y, 0, 38}], x];
CoefficientList[(f[x] - x)/x^4, x]
(* Second program: *)
a[n_] := Sum[Binomial[n-2j-1, n-3j-1] Binomial[n+3+j, n+2]/(n+3), {j, 0, (n-1)/3}]; Array[a, 35] (* Jean-François Alcover, Dec 08 2018, after David Callan *)
Table[HypergeometricPFQ[{1/3 - n/3, 2/3 - n/3, 1 - n/3, 4 + n}, {2, 1/2 - n/2, 1 - n/2}, -27/4], {n, 1, 40}] (* Vaclav Kotesovec, Sep 16 2023 *)

a(n) = Sum_{j=0..(n-1)/3} binomial[n-2j-1, n-3j-1] binomial[n+3+j, n+2]/(n+3). This counts the polygon dissections above by number j of diagonals. - David Callan, Jul 15 2004

More terms from Joerg Arndt, Jan 28 2014

A101785 G.f. satisfies: A(x) = 1 + xA(x)/(1 - x^2A(x)^2).

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 30, 79, 213, 584, 1628, 4600, 13138, 37871, 110043, 321978, 947813, 2805104, 8341608, 24912004, 74686460, 224694128, 678143656, 2052640752, 6229616730, 18952875247, 57792705415, 176596786934, 540679385663

Offset: 0

Paul D. Hanna, Dec 16 2004

nonn

Formula may be derived using the Lagrange Inversion theorem (cf. A049124).
a(n) = number of Dyck n-paths (A000108) all of whose descents have odd length. For example, a(3) counts UUUDDD, UDUDUD. - David Callan, Jul 25 2005
The number of noncrossing partitions of [n] with all blocks of odd size. E.g.: a(4)=5 with the five partitions being 123/4, 124/3, 134/2,1/234 and 1/2/3/4. - Louis Shapiro, Jan 07 2006
Number of ordered trees with n edges in which every non-leaf vertex has an odd number of children. - David Callan, Mar 30 2007
Number of valid hook configurations of permutations of [n] that avoid the patterns 312 and 321. - Colin Defant, Apr 28 2019

			Generated from Fibonacci polynomials (A011973) and
coefficients of odd powers of 1/(1-x):
a(1) = 1*1/1
a(2) = 1*1/1 + 0*1/3
a(3) = 1*1/1 + 1*3/3
a(4) = 1*1/1 + 2*6/3 + 0*1/5
a(5) = 1*1/1 + 3*10/3 + 1*5/5
a(6) = 1*1/1 + 4*15/3 + 3*15/5 + 0*1/7
a(7) = 1*1/1 + 5*21/3 + 6*35/5 + 1*7/7
a(8) = 1*1/1 + 6*28/3 + 10*70/5 + 4*28/7 + 0*1/9
This process is equivalent to the formula:
a(n) = Sum_{k=0..[(n-1)/2]} C(n-k-1,k)*C(n,2*k)/(2*k+1).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Colin Defant, Motzkin intervals and valid hook configurations, arXiv preprint arXiv:1904.10451 [math.CO], 2019.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Vaclav Kotesovec, Asymptotic of subsequences of A212382
Helmut Prodinger, Partial Dyck path interpretation for three sequences in the Encyclopedia of Integer Sequences, arXiv:2408.01290 [math.CO], 2024.

Cf. A011973, A049124.

Column k=2 of A212382.

[n eq 0 select 1 else (&+[Binomial(n-k-1,k)*Binomial(n, 2*k)/(2*k+1): k in [0..Floor((n-1)/2)]]): n in [0..30]]; // G. C. Greubel, May 03 2019

Flatten[{1,Table[Sum[Binomial[n-k-1,k]*Binomial[n,2*k]/(2*k+1),{k,0,Floor[(n-1)/2]}],{n,1,20}]}] (* Vaclav Kotesovec, Sep 17 2013 *)
CoefficientList[InverseSeries[Series[x*(1-x^2)/(1+x-x^2), {x, 0, 30}], x]/x, x] (* G. C. Greubel, May 03 2019 *)

{a(n)=if(n==0,1,sum(k=0,(n-1)\2,binomial(n-k-1,k)*binomial(n,2*k)/(2*k+1)))}
for(n=1, 40, print1(a(n), ", "))

N=66; Vec(serreverse(x/(1+sum(k=1,N,x^(2*k-1)))+O(x^N))/x) /* Joerg Arndt, Aug 19 2012 */

[1]+[sum(binomial(n-k-1, k)*binomial(n, 2*k)/(2*k+1) for k in (0..floor((n-1)/2))) for n in (1..30)] # G. C. Greubel, May 03 2019

a(n) = Sum_{k=0..[(n-1)/2]} C(n-k-1, k)*C(n, 2*k)/(2*k+1) for n>0, with a(0)=1.
G.f.: (1/x) * Series_Reversion( x*(1-x^2)/(1+x-x^2) ).
Recurrence: 4*n*(n+1)*(91*n^2 - 379*n + 360)*a(n) = 6*n*(182*n^3 - 849*n^2 + 1075*n - 264)*a(n-1) - 2*(182*n^4 - 1122*n^3 + 2011*n^2 - 603*n - 648)*a(n-2) + 6*(n-3)*(364*n^3 - 1698*n^2 + 2267*n - 696)*a(n-3) - 5*(n-4)*(n-3)*(91*n^2 - 197*n + 72)*a(n-4). - Vaclav Kotesovec, Sep 17 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 3/4 + 1/(4*sqrt(3/(19 - 304/(4103 + 273*sqrt(273))^(1/3) + 2*(4103 + 273*sqrt(273))^(1/3)))) + 1/2*sqrt(19/6 + 76/(3*(4103 + 273*sqrt(273))^(1/3)) - 1/6*(4103 + 273*sqrt(273))^(1/3) + 63/2*sqrt(3/(19 - 304/(4103 + 273*sqrt(273))^(1/3) + 2*(4103 + 273*sqrt(273))^(1/3)))) = 3.228704951094501729... is the root of the equation 5 - 24*d + 4*d^2 - 12*d^3 + 4*d^4 = 0 and c = 0.82499074317860885542266460957609663272... is the root of the equation -125 - 3376*c^2 - 22080*c^4 - 23296*c^6 + 93184*c^8 = 0. - Vaclav Kotesovec, added Sep 17 2013, updated Jan 04 2014
G.f.: 1/(9*(3-3*x+x^2))*(x^2+27- x^2*(2*x+3)^3*(x-6)^3/(9*(3-3*x+x^2)^3*S(0) - x^2*(2*x+3)^2*(x-6)^2 )), where S(k) = 4*k+3 - x^2*(2*x^2-9*x-18)^2*(3*k+4)*(6*k+5)/( 18*(4*k+5)*(3-3*x+x^2)^3 - x^2*(2*x^2-9*x-18)^2*(3*k+5)*(6*k+7)/S(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 26 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

A307413 G.f. A(x) satisfies: A(x) = 1 + xA(x)/(1 - xA(x) - 2x^2A(x)^2).

Original entry on oeis.org

1, 1, 2, 7, 26, 102, 420, 1787, 7794, 34666, 156636, 716982, 3317700, 15494156, 72935624, 345701843, 1648489762, 7902956738, 38067806892, 184152092450, 894259126540, 4357738501844, 21302682030328, 104439435098718, 513390992000340, 2529846489669412, 12494572784556440

Offset: 0

Ilya Gutkovskiy, Apr 07 2019

nonn

Unsigned version of A326564. - Paul D. Hanna, Aug 28 2019

Cf. A001045, A049124, A307411, A307412.

Cf. A326564.

terms = 26; A[] = 0; Do[A[x] = 1 + x A[x]/(1 - x A[x] - 2 x^2 A[x]^2) + O[x]^(terms + 1) // Normal, {terms + 1}]; CoefficientList[A[x], x]
terms = 27; A[] = 0; Do[A[x] = 1 + Sum[(1/3) (2^k - (-1)^k) x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 27; CoefficientList[1/x InverseSeries[Series[x (1 + x) (1 - 2 x)/(1 - 2 x^2), {x, 0, terms}], x], x]

G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} Jacobsthal(k)*x^k*A(x)^k, where Jacobsthal = A001045.
G.f.: A(x) = (1/x)*Series_Reversion(x*(1 + x)*(1 - 2*x)/(1 - 2*x^2)).
From Paul D. Hanna, Aug 29 2019: (Start)
G.f. A(x) satisfies: 0 = Sum_{n>=1} (1-(-1)^n - 2*A(x))^n * x^n / n.
G.f. A(x) satisfies: log(1 - 4*x^2*A(x)^2)/2 = arctanh(2*x - 2*x*A(x)). (End)

A307411 G.f. A(x) satisfies: A(x) = 1 + xA(x)(1 + 2xA(x))/(1 - xA(x) - x^2A(x)^2).

Original entry on oeis.org

1, 1, 4, 14, 60, 267, 1254, 6071, 30156, 152714, 785682, 4094752, 21573258, 114709363, 614777462, 3317589966, 18011350796, 98307220409, 539121535194, 2969177051678, 16415395615190, 91070109305056, 506843759000184, 2828968117483929, 15831944500607010, 88818114923080102

Offset: 0

Ilya Gutkovskiy, Apr 07 2019

nonn

Cf. A000204, A049124, A307412, A307413.

terms = 25; A[] = 0; Do[A[x] = 1 + x A[x] (1 + 2 x A[x])/(1 - x A[x] - x^2 A[x]^2) + O[x]^(terms + 1) // Normal, {terms + 1}]; CoefficientList[A[x], x]
terms = 26; A[] = 0; Do[A[x] = 1 + Sum[LucasL[k] x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 26; CoefficientList[1/x InverseSeries[Series[x (1 - x - x^2)/(1 + x^2), {x, 0, terms}], x], x]

G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} Lucas(k)*x^k*A(x)^k, where Lucas = A000204.

G.f.: A(x) = (1/x)*Series_Reversion(x*(1 - x - x^2)/(1 + x^2)).

A307412 G.f. A(x) satisfies: A(x) = 1 + xA(x)/(1 - 2xA(x) - x^2A(x)^2).

Original entry on oeis.org

1, 1, 3, 12, 53, 250, 1234, 6295, 32925, 175616, 951596, 5223658, 28987546, 162349759, 916502869, 5209630108, 29792226533, 171284524184, 989460348216, 5740230703588, 33429379234924, 195361236443008, 1145312096390408, 6733896357727242, 39697441350016170, 234596104853541967

Offset: 0

Ilya Gutkovskiy, Apr 07 2019

nonn

Cf. A000129, A049124, A307411, A307413.

terms = 25; A[] = 0; Do[A[x] = 1 + x A[x]/(1 - 2 x A[x] - x^2 A[x]^2) + O[x]^(terms + 1) // Normal, {terms + 1}]; CoefficientList[A[x], x]
terms = 26; A[] = 0; Do[A[x] = 1 + Sum[Fibonacci[k, 2] x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 26; CoefficientList[1/x InverseSeries[Series[x (1 - 2 x - x^2)/(1 - x - x^2), {x, 0, terms}], x], x]

G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} Pell(k)*x^k*A(x)^k, where Pell = A000129.
G.f.: A(x) = (1/x)*Series_Reversion(x*(1 - 2*x - x^2)/(1 - x - x^2)).
a(n) ~ sqrt((1 + 2^(1/3))*(4 + 7*2^(1/3))) * (2 + 3/2^(2/3) + 3/2^(1/3))^n / (3 * sqrt(Pi) * (2*n)^(3/2)). - Vaclav Kotesovec, Nov 05 2021

A269228 Number of nondirected diagonally convex polyominoes with perimeter 2n + 2.

Original entry on oeis.org

1, 2, 7, 28, 122, 556, 2618, 12634, 62128, 310212, 1568495, 8014742, 41323641, 214719610, 1123244757, 5910863420, 31268459118, 166185855552, 886961294034, 4751819567488, 25545030878475, 137756210983218, 745003421378887, 4039670554117446, 21957581725458521

Offset: 1

Svjetlan Feretic, Jul 11 2016

nonn

The generating function satisfies an algebraic equation of degree eight. I computed that generating function using the "turbo Temperley" method.
The formula for the generating function is given in the enclosed Maple worksheet.
The most practical version of the "turbo Temperley" method was given in Bousquet-Mélou's paper cited below.
The first five terms are the same as in the sequence A005435.
A005435(n) is the number of column-convex polyominoes with perimeter 2n + 2.
A049124(n) is the number of directed diagonally convex polyominoes with perimeter 2n.

			a(7) = 2618, so there are 2618 nondirected diagonally convex polyominoes with perimeter 2*7 + 2 = 16.

Svjetlan Feretic, Table of n, a(n) for n = 1..100
M. Bousquet-Mélou, A method for the enumeration of various classes of column-convex polygons, Discrete Math. 154 (1996), 1-25.
Svjetlan Feretić, Maple worksheet with g.f.
Svjetlan Feretić, the first one hundred terms of the sequence A269228
Svjetlan Feretić, The perimeter generating function for nondirected diagonally convex polyominoes, arXiv:1907.09409 [math.CO], 2019.

Cf. A005435, A049124.

A290816 Number of dissections of an n-gon into polygons with odd number of sides counted up to rotations and reflections.

Original entry on oeis.org

1, 1, 2, 4, 8, 23, 65, 223, 757, 2824, 10559, 40994, 160734, 641420, 2584587, 10528305, 43237978, 178974779, 745814185, 3127246179, 13185588894, 55878618492, 237905685582, 1017225981255, 4366536472758, 18812074137141, 81320795918871, 352638701880227

Offset: 3

Evgeniy Krasko, Sep 03 2017

nonn

			For a(5) = 2 the dissections of a pentagon are: a dissection into 3 triangles; a dissection into one pentagon.

Andrew Howroyd, Table of n, a(n) for n = 3..200
E. Krasko, A. Omelchenko, Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.

Cf. A049124 (counted distinctly).

Cf. A001004, A290722, A295419.

(* See A295419 for DissectionsModDihedral *)
DissectionsModDihedral[Mod[#, 2]& /@ Range[1, 31]] (* Jean-François Alcover, Sep 25 2019, after Andrew Howroyd *)

\\ See A295419 for DissectionsModDihedral().
DissectionsModDihedral(apply(v->v%2, [1..25])) \\ Andrew Howroyd, Nov 22 2017

Terms a(16) and beyond from Andrew Howroyd, Nov 22 2017

A364833 G.f. satisfies A(x) = 1 + xA(x)^2/(1 - x^3A(x)^3).

Original entry on oeis.org

1, 1, 2, 5, 15, 49, 168, 595, 2160, 7997, 30083, 114660, 441840, 1718531, 6737820, 26600784, 105659970, 421949492, 1693120779, 6823018035, 27602090087, 112053680381, 456343848121, 1863893501065, 7633232165286, 31337360839387, 128944120202510

Offset: 0

Seiichi Manyama, Aug 28 2023

nonn

Cf. A112806, A219537.
Cf. A218251, A364161, A365247.
Cf. A001003, A049124.

A364833 := proc(n)
    add( binomial(n-2*k-1,k)*binomial(2*n-3*k+1,n-3*k)/ (2*n-3*k+1),k=0..floor(n/3)) ;
end proc:
seq(A364833(n),n=0..80); # R. J. Mathar, Aug 29 2023

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

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k) * binomial(2*n-3*k+1,n-3*k)/(2*n-3*k+1).
D-finite with recurrence 31*n*(626109182191*n-1858292669035) *(n-1)*(n+1) *a(n) -n*(n-1) *(244150473843619*n^2 -1454194662255591*n +2175006457069082) *a(n-1) +3*(n-1) *(292927551362415*n^3 -2593205532882651*n^2 +7084566217454162*n -5823331737745632)*a(n-2) +(-843955616916167*n^4 +9932491073296715*n^3 -42016891739306929*n^2 +76184884157722453*n -50166914106142776) *a(n-3) +18*(1509721335071*n^4 -40413442328880*n^3 +330301781039401*n^2 -1078322794857576*n +1231650372542192) *a(n-4) +18*(39673125909769*n^4 -598320530478001*n^3 +3228489073613917*n^2 -7321259523567459*n +5788776339353646) *a(n-5) +27*(n-5) *(3102413205331*n^3 -35996479327373*n^2 +114122791959960*n -64735736097804) *a(n-6) -243*(n-6) *(n-7)*(475638134099*n^2 -2399948859181*n +2877042451214) *a(n-7) -243*(45857481910*n -35520400961) *(n-5) *(n-7) *(n-8)*a(n-8)=0. - R. J. Mathar, Aug 29 2023
G.f.: (1/x) * Series_Reversion( x*(1 - x / (1 - x^3)) ). - Seiichi Manyama, Sep 28 2024

cp's OEIS Frontend

A003168 Number of blobs with 2n+1 edges.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

Formula

A054514 Number of ways to place non-crossing diagonals in convex (n+4)-gon so as to create no triangles or quadrilaterals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A101785 G.f. satisfies: A(x) = 1 + x*A(x)/(1 - x^2*A(x)^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

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

A307413 G.f. A(x) satisfies: A(x) = 1 + x*A(x)/(1 - x*A(x) - 2*x^2*A(x)^2).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A307411 G.f. A(x) satisfies: A(x) = 1 + x*A(x)*(1 + 2*x*A(x))/(1 - x*A(x) - x^2*A(x)^2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A307412 G.f. A(x) satisfies: A(x) = 1 + x*A(x)/(1 - 2*x*A(x) - x^2*A(x)^2).

Views

Author

Keywords

Crossrefs

A101785 G.f. satisfies: A(x) = 1 + xA(x)/(1 - x^2A(x)^2).

A307413 G.f. A(x) satisfies: A(x) = 1 + xA(x)/(1 - xA(x) - 2x^2A(x)^2).

A307411 G.f. A(x) satisfies: A(x) = 1 + xA(x)(1 + 2xA(x))/(1 - xA(x) - x^2A(x)^2).

A307412 G.f. A(x) satisfies: A(x) = 1 + xA(x)/(1 - 2xA(x) - x^2A(x)^2).

A364833 G.f. satisfies A(x) = 1 + xA(x)^2/(1 - x^3A(x)^3).