A243662 -id:A243662 - 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

A102537 Triangle T(n,k) read by rows: (1/n) * C(2n+k,k-1) * C(n,k); n, k >= 1.

Original entry on oeis.org

1, 1, 3, 1, 8, 12, 1, 15, 55, 55, 1, 24, 156, 364, 273, 1, 35, 350, 1400, 2380, 1428, 1, 48, 680, 4080, 11628, 15504, 7752, 1, 63, 1197, 9975, 41895, 92169, 100947, 43263, 1, 80, 1960, 21560, 123970, 396704, 708400, 657800, 246675, 1, 99, 3036, 42504

Offset: 1

Ralf Stephan, Jan 14 2005

Number of dissections of a convex (2n+2)-gon by k-1 noncrossing diagonals into (2j+2)-gons, 1 <= j <= n-1.
Apparently, a signed, refined version of this array is given on page 65 of the Einziger link, related to the antipode of a Hopf algebra. - Tom Copeland, May 19 2015
The f-vectors of the simplicial noncrossing hypertree complexes of McCammond (p. 15). The reduced Euler characteristics are the signed Catalan numbers A000108. - Tom Copeland, May 19 2017
The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial ((x+1)*(x+2)*...*(x+2n+1))*((x+n+2)*(x+n+3)*...*(x+2n)) / ((2n+1)!*(n)!) in the basis made of the binomial(x+i,i). - F. Chapoton, Nov 01 2022
Chapoton's observation above is correct: the precise expansion is ((x+1)*(x+2)*...*(x+2n+1))*((x+n+2)*(x+n+3)*...*(x+2n)) / ((2n+1)!*n!) = Sum_{k = 1..n} (-1)^(k+1)*T(n,n+1-k)*binomial(x+3*n+1-k, 3*n+1-k), as can be verified using the WZ algorithm. For example, n = 3 gives (x+1)*(x+2)*(x+3)*(x+4)*(x+5)*(x+6)*(x+7)*(x+5)(x+6)/(7!*3!) = 12*binomial(x+9,9) - 8*binomial(x+8,8) + binomial(x+7,7). - Peter Bala, Jun 25 2023

			Triangle begins
  1;
  1,  3;
  1,  8,   12;
  1, 15,   55,    55;
  1, 24,  156,   364,    273;
  1, 35,  350,  1400,   2380,   1428;
  1, 48,  680,  4080,  11628,  15504,   7752;
  1, 63, 1197,  9975,  41895,  92169, 100947,  43263;
  1, 80, 1960, 21560, 123970, 396704, 708400, 657800, 246675;

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150).
H. Einziger, Incidence Hopf algebras: Antipodes, forest formulas, and noncrossing partitions, Dissertation (2010), George Washington University.
J. McCammond, Noncrossing Hypertrees, 2015.
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.
E. Tzanaki, Polygon dissections and some generalizations of cluster complexes, arXiv:math/0501100 [math.CO], 2005.

Left-hand columns include A005563. Right-hand columns include essentially A001764 and A013698.
Row sums are in A003168.
Cf. A000108, A120986.
Cf. A243662 for rows reversed.

[[1/n * Binomial(2*n+k,k-1) * Binomial(n,k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, May 20 2015

Table[1/n*Binomial[2 n + k, k - 1] Binomial[n, k], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, May 20 2017 *)

A243663 Triangle read by rows: the reversed x = 1+q Narayana triangle at m=3.

Original entry on oeis.org

1, 4, 1, 22, 11, 1, 140, 105, 21, 1, 969, 969, 306, 34, 1, 7084, 8855, 3850, 700, 50, 1, 53820, 80730, 44850, 11500, 1380, 69, 1, 420732, 736281, 498771, 166257, 28665, 2457, 91, 1, 3362260, 6724520, 5379616, 2215136, 503440, 62930, 4060, 116, 1

Offset: 1

N. J. A. Sloane, Jun 13 2014

See Novelli-Thibon (2014) for precise definition.

			Triangle begins:
     1
     4,    1
    22,   11,    1
   140,  105,   21,   1
   969,  969,  306,  34,  1
  7084, 8855, 3850, 700, 50, 1
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962, 2014. See Fig. 11.

Cf. A001263, A243662 (m=2).

T[m_][n_, k_] := Binomial[(m + 1) n + 1 - k, n - k] Binomial[n, k - 1]/n;
Table[T[3][n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 12 2019 *)

T(n,k) = binomial(4*n+1-k,n-k) * binomial(n,k-1) / n for 1 <= k <= n, more generally: T_m(n,k) = binomial((m+1)*n+1-k,n-k) * binomial(n,k-1) / n for 1 <= k <= n and some fixed integer m > 1. - Werner Schulte, Nov 22 2018

More terms from Werner Schulte, Nov 22 2018

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A003168 Number of blobs with 2n+1 edges.

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A102537 Triangle T(n,k) read by rows: (1/n) * C(2n+k,k-1) * C(n,k); n, k >= 1.

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A243663 Triangle read by rows: the reversed x = 1+q Narayana triangle at m=3.

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