A006013 -id:A006013 - cp's OEIS frontend

A102893 Number of noncrossing trees with n edges and having degree of the root at least 2.

1, 0, 1, 5, 25, 130, 700, 3876, 21945, 126500, 740025, 4382625, 26225628, 158331880, 963250600, 5899491640, 36345082425, 225082957512, 1400431689475, 8749779798375, 54874635255825, 345329274848250, 2179969531405680

Offset: 0

Emeric Deutsch, Jan 16 2005

[a(n+2)]= [1,5,25,130,700,...] is the self-convolution 5th power of A001764. - Philippe Deléham, Nov 11 2009

a(n) is the number of dissections of a convex (2n+2)-sided polygon by nonintersecting diagonals into quadrilaterals such that at least one of the dividing diagonals passes through a chosen vertex. - Muhammed Sefa Saydam, Jan 24 2025

			a(2)=1 because among the noncrossing trees with 2 edges, namely /_, _\ and /\, only the last one has root degree >1.

Andrew Howroyd, Table of n, a(n) for n = 0..200
David Bevan, Robert Brignall, Andrew Elvey Price and Jay Pantone, A structural characterisation of Av(1324) and new bounds on its growth rate, arXiv preprint arXiv:1711.10325 [math.CO], 2017-2019.
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.

Column k=0 of A102892 and column k=0 of A102593.

Cf. A001764, A006013.

a:=proc(n) if n=0 then 1 else 5*binomial(3*n-1,n-2)/(3*n-1) fi end:
seq(a(n), n=0..25);
# Recurrence:
a := proc(n) option remember; if n < 3 then return [1,0,1][n+1] fi;
(27*n^3 - 81*n^2 + 78*n - 24)*a(n - 1)/(4*n^3 - 6*n^2 - 4*n) end:
seq(a(n), n=0..23); # Peter Luschny, Aug 08 2020
alias(PS=ListTools:-PartialSums): A102893List := proc(m) local A, P, n;
A := [1,0]; P := [1]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));
A := [op(A), P[-2]] od; A end: A102893List(23); # Peter Luschny, Mar 26 2022

a[0] = 1; a[n_] := 5*Binomial[3n-1, n-2]/(3n-1); Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Mar 01 2018 *)

a(n) = if(n<=1, n==0, 5*binomial(3*n-1, n-2)/(3*n-1)); \\ Andrew Howroyd, Nov 17 2017

a(0)=1; a(n) = 5*binomial(3n-1, n-2)/(3n-1) if n > 0.
G.f.: g - z*g^2, where g = 1 + z*g^3 is the g.f. of the ternary numbers (A001764).
a(n) = A001764(n) - A006013(n-1).
D-finite with recurrence 2*n*(2*n+1)*(n-2)*a(n) -3*(n-1)*(3*n-4)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Feb 16 2018
a(n) ~ (5*3^(3*n + 1/2))/(36*4^n*n^(3/2)*sqrt(Pi)). - Peter Luschny, Aug 08 2020

A143363 Number of ordered trees with n edges and having no protected vertices. A protected vertex in an ordered tree is a vertex at least 2 edges away from its leaf descendants.

Original entry on oeis.org

1, 1, 1, 3, 6, 17, 43, 123, 343, 1004, 2938, 8791, 26456, 80597, 247091, 763507, 2372334, 7413119, 23271657, 73376140, 232238350, 737638868, 2350318688, 7510620143, 24064672921, 77294975952, 248832007318, 802737926643

Offset: 0

Emeric Deutsch, Aug 20 2008

nonn

The "no protected vertices" condition can be rephrased as "every non-leaf vertex has at least one leaf child". But a(n) is also the number of ordered trees with n edges in which every non-leaf vertex has at most one leaf child. - David Callan, Aug 22 2014
Also the number of locally non-intersecting ordered rooted trees with n edges, meaning every non-leaf subtree has empty intersection. The unordered version is A007562. - Gus Wiseman, Nov 19 2022
a(n) is the number of parking functions of size n-1 avoiding the patterns 123, 132, and 213 . - Lara Pudwell, Apr 10 2023
For n>0, a(n) is the number of ways to place non-intersecting diagonals in convex n+3-gon so as to create no triangles such that none of the dividing diagonals passes through a chosen vertex. (empirical observation) - Muhammed Sefa Saydam, Feb 14 2025 and Aug 05 2025

			From _Gus Wiseman_, Nov 19 2022: (Start)
The a(0) = 1 through a(4) = 6 trees with at least one leaf directly under any non-leaf node:
  o  (o)  (oo)  (ooo)   (oooo)
                ((o)o)  ((o)oo)
                (o(o))  ((oo)o)
                        (o(o)o)
                        (o(oo))
                        (oo(o))
The a(0) = 1 through a(4) = 6 trees with at most one leaf directly under any node:
  o  (o)  ((o))  ((o)o)   (((o))o)
                 (o(o))   (((o)o))
                 (((o)))  ((o)(o))
                          ((o(o)))
                          (o((o)))
                          ((((o))))
(End)

Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
Gi-Sang Cheon and Louis W. Shapiro, Protected points in ordered trees, Appl. Math. Letters, 21, 2008, 516-520.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016.

Cf. A143362.
For exactly one leaf directly under any node we have A006013.
The unordered version is A007562, ranked by A316470.
Allowing lone children gives A319378.
A000108 counts ordered rooted trees, unordered A000081.
A358453 counts transitive ordered trees, unordered A290689.
A358460 counts locally disjoint ordered trees, unordered A316473.
Cf. A318185, A324756, A324767, A324840, A324844.

p:=z^2*G^3-2*z*G^2-2*z^2*G^2+3*z*G+G+z^2*G-1-2*z=0: G:=RootOf(p,G): Gser:= series(G,z=0,33): seq(coeff(Gser,z,n),n=0..28);

a[n_Integer] := a[n] = Round[SeriesCoefficient[2 (x + 1 - Sqrt[x^2 - x + 1] Cos[ArcTan[(3 x Sqrt[12 x^3 - 96 x^2 - 24 x + 15])/(2 x^3 - 30 x^2 - 3 x + 2)]/3])/(3 x), {x, 0, n}]]; Table[a[n], {n, 0, 20}] (* Vladimir Reshetnikov, Apr 10 2022 *)
RecurrenceTable[{25 (n + 5) (n + 6) a[n + 5] - 10 (n + 5) (5 n + 21) a[n + 4] - 2 (77 n^2 + 613 n + 1185) a[n + 3] + 2 (50 n^2 + 253 n + 312) a[n + 2] + 4 (2 n + 1) (7 n + 9) a[n + 1] - 4 n (2 n + 1) a[n] == 0, a[0] == 1, a[1] == 1, a[2] == 1, a[3] == 3, a[4] == 6}, a[n], {n, 0, 27}] (* Vladimir Reshetnikov, Apr 11 2022 *)
ait[n_]:=ait[n]=If[n==1,{{}},Join@@Table[Select[Tuples[ait/@c],MemberQ[#,{}]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[ait[n]],{n,15}] (* Gus Wiseman, Nov 19 2022 *)

a(n) = A143362(n,0) for n>=1.
G.f.: G=G(z) satisfies z^2*G^3-2z(1+z)G^2+(1+3z+z^2)G-(1+2z)=0.
G.f.: (x+1-sqrt(x^2-x+1)*cos(arctan((3*x*sqrt(12*x^3-96*x^2-24*x+15))/(2*x^3-30*x^2-3*x+2))/3))*2/(3*x). - Vladimir Reshetnikov, Apr 10 2022
Recurrence: 25*(n+5)*(n+6)*a(n+5) - 10*(n+5)*(5*n+21)*a(n+4) - 2*(77*n^2+613*n+1185)*a(n+3) + 2*(50*n^2+253*n+312)*a(n+2) + 4*(2*n+1)*(7*n+9)*a(n+1) - 4*n*(2*n+1)*a(n) = 0. - Vladimir Reshetnikov, Apr 11 2022
From Muhammed Sefa Saydam, Jul 12 2025: (Start)
a(n) =  Sum_{k=2..n+2} A046736(k) * A046736(n-k+3) , for n >= 0 and A046736(1) = 1.
a(n) = A049125(n) + Sum_{k=1..n-2} A049125(k) * A046736(n-k+2), for n >= 3.
a(n) = A049125(n) + Sum_{k=1..n-2} a(k) * a(n-k-1), for n >= 3. (End)

A063030 Reversion of y - y^2 - y^4 + y^5.

Original entry on oeis.org

0, 1, 1, 2, 6, 19, 63, 220, 795, 2942, 11099, 42536, 165126, 647955, 2565946, 10241616, 41158598, 166402323, 676338003, 2761988994, 11327162406, 46631572295, 192638451780, 798316442580, 3317866307145, 13825837134096

Offset: 0

Olivier Gérard, Jul 05 2001

R. J. Mathar, Table of n, a(n) for n = 0..105
Index entries for reversions of series

Cf. A006013, A063020.

Cf. A063026.

CoefficientList[InverseSeries[Series[y - y^2 - y^4 + y^5, {y, 0, 30}], x], x]

a(n)=if(n<1,0,polcoeff(serreverse(x-x^2-x^4+x^5+x*O(x^n)),n))

D-finite with recurrence 1458*n*(n-1)*(n-2)*(2*n-1) *(981649511*n -2631216939)*a(n) -486*(n-1)*(n-2) *(24210415932*n^3 -114067288649*n^2 +155533650884*n -64732315335)*a(n-1) +54*(n-2) *(39787015892*n^4 -313539301751*n^3 +992577496688*n^2 -1613867842189*n +1173502139880)*a(n-2) +(-27607572942679*n^5 +295135536608825*n^4 -1205223186688595*n^3 +2314131935158975*n^2 -2033367943220766*n +619177732684560)*a(n-3) -5*(5*n-21) *(5408009*n +1144402484)*(5*n-19) *(5*n-18)*(5*n-17) *a(n-4)=0. - R. J. Mathar, Mar 21 2022

a(n+1) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(n+k,n) * binomial(2*n-3*k,n). - Seiichi Manyama, Sep 26 2023

A278880 Triangle where g.f. S = S(x,m) satisfies: S = x/(G(-S^2)G(-mS^2)) such that G(x) = 1 + xG(x)^2 is the g.f. of the Catalan numbers (A000108), as read by rows of coefficients T(n,k) of x^(2n-1)*m^k in S(x,m) for n>=1, k=0..n-1.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 14, 14, 1, 1, 30, 81, 30, 1, 1, 55, 308, 308, 55, 1, 1, 91, 910, 1872, 910, 91, 1, 1, 140, 2268, 8250, 8250, 2268, 140, 1, 1, 204, 4998, 29172, 51425, 29172, 4998, 204, 1, 1, 285, 10032, 87780, 247247, 247247, 87780, 10032, 285, 1, 1, 385, 18711, 233376, 980980, 1565109, 980980, 233376, 18711, 385, 1, 1, 506, 32890, 562419, 3354780, 7970144, 7970144, 3354780, 562419, 32890, 506, 1

Offset: 1

Paul D. Hanna, Nov 29 2016

T(n,k) = the number of fighting fish with (n-k) left lower free and (k+1) right lower free edges with a marked tail. [See Theorem 3 in the Duchi reference on Fighting Fish: enumerative properties.] - Paul D. Hanna, Dec 08 2016

			This triangle of coefficients of x^(2*n-1)*m^k in S(x,m) for n>=1, k=0..n-1, begins:
  1;
  1, 1;
  1, 5, 1;
  1, 14, 14, 1;
  1, 30, 81, 30, 1;
  1, 55, 308, 308, 55, 1;
  1, 91, 910, 1872, 910, 91, 1;
  1, 140, 2268, 8250, 8250, 2268, 140, 1;
  1, 204, 4998, 29172, 51425, 29172, 4998, 204, 1;
  1, 285, 10032, 87780, 247247, 247247, 87780, 10032, 285, 1;
  1, 385, 18711, 233376, 980980, 1565109, 980980, 233376, 18711, 385, 1;
  1, 506, 32890, 562419, 3354780, 7970144, 7970144, 3354780, 562419, 32890, 506, 1; ...
Generating function:
S(x,m) = x + (m + 1)*x^3 + (m^2 + 5*m + 1)*x^5 +
 (m^3 + 14*m^2 + 14*m + 1)*x^7 +
 (m^4 + 30*m^3 + 81*m^2 + 30*m + 1)*x^9 +
 (m^5 + 55*m^4 + 308*m^3 + 308*m^2 + 55*m + 1)*x^11 +
 (m^6 + 91*m^5 + 910*m^4 + 1872*m^3 + 910*m^2 + 91*m + 1)*x^13 +
 (m^7 + 140*m^6 + 2268*m^5 + 8250*m^4 + 8250*m^3 + 2268*m^2 + 140*m + 1)*x^15 +...
where S = S(x,m) satisfies:
S = x / ( G(-S^2) * G(-m*S^2) ) such that G(x) = 1 + x*G(x)^2.
Also,
S = x * (1 + x*S) * (1 + m*x*S) / (1 - m*x^2*S^2)^2,
where related series C = C(x,m) and D = D(x,m) satisfy
S = x*C*D, C = 1 + x*S*D, and D = 1 + m*x*S*C,
such that
C = C^2 - S^2,
D = D^2 - m*S^2.
...
The square of the g.f. begins:
S(x,m)^2 = x^2 + (2*m + 2)*x^4 + (3*m^2 + 12*m + 3)*x^6 +
 (4*m^3 + 40*m^2 + 40*m + 4)*x^8 + (5*m^4 + 100*m^3 + 245*m^2 + 100*m + 5)*x^10 +
 (6*m^5 + 210*m^4 + 1008*m^3 + 1008*m^2 + 210*m + 6)*x^12 +
 (7*m^6 + 392*m^5 + 3234*m^4 + 6300*m^3 + 3234*m^2 + 392*m + 7)*x^14 +
 (8*m^7 + 672*m^6 + 8736*m^5 + 29040*m^4 + 29040*m^3 + 8736*m^2 + 672*m + 8)*x^16 +...+ x^(2*n)*Sum_{k=0,n-1} n*A082680(n,k+1)*m^k +...
where A082680(n,k+1) = (n+k)!*(2*n-k-1)!/((k+1)!*(n-k)!*(2*k+1)!*(2*n-2*k-1)!).

Paul D. Hanna, Table of n, a(n) for n = 1..1035 for rows 1..45 of the flattened form of this triangle.
Enrica Duchi, Veronica Guerrini, Simone Rinaldi, and Gilles Schaeffer, Fighting Fish: enumerative properties, arXiv:1611.04625 [math.CO], 2016.
Thomas Einolf, Robert Muth, and Jeffrey Wilkinson, Injectively k-colored rooted forests, arXiv:2107.13417 [math.CO], 2021.
Zai-Ting Huang, Congruence Properties From Fish Numbers, Master's thesis, Nat'l Taiwan Normal Univ. (2025) 31944792.

Cf. A278881 (C(x,m)), A278882 (D(x,m)), A278883 (central terms).

Cf. A000108, A006013 (row sums), A258313, A278745, A082680.

T[n_, k_] := (2n-1)/((2n-2k-1)(2k+1)) Binomial[2n-k-2, k] Binomial[n+k-1, n-k-1];
Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 26 2018 *)

{T(n,k) = my(S=x,C=1,D=1); for(i=0,2*n, S = x*C*D + O(x^(2*n+2)); C = 1 + x*S*D; D = 1 + m*x*S*C;); polcoeff(polcoeff(S,2*n-1,x),k,m)}
for(n=1,15, for(k=0,n-1, print1(T(n,k),", "));print(""))

/* Explicit formula for T(n,k) */
{T(n,k) = (2*n-1)/((2*n-2*k-1)*(2*k+1)) * binomial(2*n-k-2,k) * binomial(n+k-1,n-k-1)}
for(n=1,15, for(k=0,n-1, print1(T(n,k),", "));print(""))

G.f. S = S(x,m), and related functions C = C(x,m) and D = D(x,m) satisfy:
(1.a) S = x*C*D.
(1.b) C = 1 + x*S*D.
(1.c) D = 1 + m*x*S*C.
...
(2.a) C = C^2 - S^2.
(2.b) D = D^2 - m*S^2.
(2.c) C = (1 + sqrt(1 + 4*S^2))/2.
(2.d) D = (1 + sqrt(1 + 4*m*S^2))/2.
...
(3.a) S = x*(1 + x*S)*(1 + m*x*S) / (1 - m*x^2*S^2)^2.
(3.b) C = (1 + x*S) / (1 - m*x^2*S^2).
(3.c) D = (1 + m*x*S) / (1 - m*x^2*S^2).
(3.d) S = x/((1 - x^2*D^2)*(1 - m*x^2*C^2)).
(3.e) C = 1/(1 - x^2*D^2).
(3.f) D = 1/(1 - m*x^2*C^2).
...
(4.a) x = m^2*x^4*S^5 - 2*m*x^2*S^3 - m*x^3*S^2 + (1 - (m+1)*x^2)*S.
(4.b) 0 = 1 - (1-x^2)*C - 2*m*x^2*C^2 + 2*m*x^2*C^3 + m^2*x^4*C^4 - m^2*x^4*C^5.
(4.c) 0 = 1 - (1-m*x^2)*D - 2*x^2*D^2 + 2*x^2*D^3 + x^4*D^4 - x^4*D^5.
...
(5.a) S(x,m) = Series_Reversion( x*G(-x^2)*G(-m*x^2) ), where G(x) = 1 + x*G(x)^2 is the g.f. of the Catalan numbers (A000108).
Logarithmic derivatives.
(6.a) C'/C = 2*S*S' / (C^2 + S^2).
(6.b) D'/D = 2*m*S*S' / (D^2 + m*S^2).
...
(7.a) S(x,m)^2 = Sum_{n>=1} x^(2*n) * Sum_{k=0,n-1} n*A082680(n,k+1)*m^k, where A082680(n,k+1) = (n+k)!*(2*n-k-1)!/((k+1)!*(n-k)!*(2*k+1)!*(2*n-2*k-1)!).
...
T(n,k) = (2*n-1)/((2*n-2*k-1)*(2*k+1)) * binomial(2*n-k-2,k) * binomial(n+k-1,n-k-1). [From Theorem 3 in the Duchi reference] - Paul D. Hanna, Dec 08 2016
Row sums yield A006013(n-1) = binomial(3*n-2,n-1)/n for n>=1.
Central terms: T(2*n+1, n) = (4*n-3) * ( binomial(3*n-3,n-1)/(2*n-1) )^2 for n>=1.
Sum_{k=0..n-1} 2^k * T(n,k) = A258313(n-1) for n>=1.
Sum_{k=0..2*n-2} (-1)^k * T(2*n-1,k) = A278745(n) for n>=1.

A286784 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 5, 15, 9, 1, 14, 56, 56, 16, 1, 42, 210, 300, 150, 25, 1, 132, 792, 1485, 1100, 330, 36, 1, 429, 3003, 7007, 7007, 3185, 637, 49, 1, 1430, 11440, 32032, 40768, 25480, 7840, 1120, 64, 1, 4862, 43758, 143208, 222768, 179928, 77112, 17136, 1836, 81, 1, 16796, 167960, 629850, 1162800, 1162800, 651168, 203490, 34200, 2850, 100, 1

Offset: 0

Gheorghe Coserea, May 14 2017

T(n,k) is the number of Feynman's diagrams with k fermionic loops in the order n of the perturbative expansion in dimension zero for the GW approximation of the self-energy function in a many-body theory of fermions with two-body interaction (see Molinari link).

			A(x;t) = 1 + (1 + t)*x + (2 + 4*t + t^2)*x^2 + (5 + 15*t + 9*t^2 + t^3)*x^3 + ...
Triangle starts:
n\k  [0]   [1]    [2]     [3]     [4]     [5]    [6]    [7]   [8] [9]
[0]  1;
[1]  1,    1;
[2]  2,    4,     1;
[3]  5,    15,    9,      1;
[4]  14,   56,    56,     16,     1;
[5]  42,   210,   300,    150,    25,     1;
[6]  132,  792,   1485,   1100,   330,    36,    1;
[7]  429,  3003,  7007,   7007,   3185,   637,   49,    1;
[8]  1430, 11440, 32032,  40768,  25480,  7840,  1120,  64,   1;
[9]  4862, 43758, 143208, 222768, 179928, 77112, 17136, 1836, 81, 1;
[10] ...

Gheorghe Coserea, Rows n=0..122, flattened
Luca G. Molinari, Hedin's equations and enumeration of Feynman's diagrams, arXiv:cond-mat/0401500 [cond-mat.str-el], 2005.

Cf. A286781, A286782, A286783.

/* As triangle */ [[(Binomial(2*n, n+m)*Binomial(n+1, m))/(n+1): m in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 23 2018

Flatten@Table[Binomial[2 n, n + m] Binomial[n + 1, m] / (n + 1), {n, 0, 10}, {m, 0, n}] (* Vincenzo Librandi, Sep 23 2018 *)

T(n,m):=(binomial(2*n,n+m)*binomial(n+1,m))/(n+1); /* Vladimir Kruchinin, Sep 23 2018 */

A286784_ser(N,t='t) = my(x='x+O('x^N)); serreverse(Ser(x*(1-x)^2/(1+(t-1)*x)))/x;
concat(apply(p->Vecrev(p), Vec(A286784_ser(12))))
\\ test: y=A286784_ser(50); y*(1-x*y)^2 == 1 + ('t-1)*x*y

y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies y*(1-x*y)^2 = 1 + (t-1)*x*y, where P_n(t) = Sum_{k=0..n} T(n,k)*t^k.
A000108(n) = T(n,0), A001791(n) = T(n,1), A002055(n+3) = T(n,2), A000290(n) = T(n,n-1), A006013(n) = P_n(1), A003169(n+1) = P_n(2).
T(n,m) = C(2*n,n+m)*C(n+1,m)/(n+1). - Vladimir Kruchinin, Sep 23 2018

A085405 Common residues of binomial(3n+2,n+1)/(3n+2) modulo 2.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul D. Hanna, Jun 29 2003

nonn

The positions of ones are given by A022340 and runs of zeros are given by A085407: both are related to the Fibonacci sequence.

Antti Karttunen, Table of n, a(n) for n = 0..65539
Index entries for characteristic functions

Cf. A005940, A006013, A022340 (ones), A085407 (zeros), A085357, A277332, A323239.

A085405(n) = ((binomial((3*n)+2, n+1)/((3*n)+2))%2); \\ Antti Karttunen, Jan 12 2019

A085405(n) = if(n%2,0,while(n>0, my(nextn=(n>>1)); if(1==(nextn%2)*(n%2), return(0)); n = nextn); (1)); \\ (Much faster than above program) - Antti Karttunen, Jan 12 2019

a(n) = C(3n+2, n+1)/(3n+2) (Mod 2) = A006013(n) (Mod 2), where A006013 is the self-convolution of A001764 (ternary trees).

a(n) = A323239(A005940(1+n)). - Antti Karttunen, Jan 12 2019

A110616 A convolution triangle of numbers based on A001764.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 12, 7, 3, 1, 55, 30, 12, 4, 1, 273, 143, 55, 18, 5, 1, 1428, 728, 273, 88, 25, 6, 1, 7752, 3876, 1428, 455, 130, 33, 7, 1, 43263, 21318, 7752, 2448, 700, 182, 42, 8, 1, 246675, 120175, 43263, 13566, 3876, 1020, 245, 52, 9, 1

Offset: 0

Philippe Deléham, Sep 14 2005, Jun 15 2007

Reflected version of A069269. - Vladeta Jovovic, Sep 27 2006
With offset 1 for n and k, T(n,k) = number of Dyck paths of semilength n for which all descents are of even length (counted by A001764) with no valley vertices at height 1 and with k returns to ground level. For example, T(3,2)=2 counts U^4 D^4 U^2 D^2, U^2 D^2 U^4 D^4 where U=upstep, D=downstep and exponents denote repetition. - David Callan, Aug 27 2009
Riordan array (f(x), x*f(x)) with f(x) = (2/sqrt(3*x))*sin((1/3)*arcsin(sqrt(27*x/4))). - Philippe Deléham, Jan 27 2014
Antidiagonals of convolution matrix of Table 1.4, p. 397, of Hoggatt and Bicknell. - Tom Copeland, Dec 25 2019

			Triangle begins:
       1;
       1,      1;
       3,      2,     1;
      12,      7,     3,     1;
      55,     30,    12,     4,    1;
     273,    143,    55,    18,    5,    1;
    1428,    728,   273,    88,   25,    6,   1;
    7752,   3876,  1428,   455,  130,   33,   7,  1;
   43263,  21318,  7752,  2448,  700,  182,  42,  8, 1;
  246675, 120175, 43263, 13566, 3876, 1020, 245, 52, 9, 1;
  ...
From _Peter Bala_, Feb 04 2025: (Start)
The transposed array factorizes as an infinite product of upper triangular arrays:
  / 1               \^T   /1             \^T /1             \^T / 1            \^T
  | 1    1           |   | 1   1          | | 0  1           |  | 0  1          |
  | 3    2   1       | = | 2   1   1      | | 0  1   1       |  | 0  0  1       | ...
  |12    7   3   1   |   | 5   2   1  1   | | 0  2   1  1    |  | 0  0  1  1    |
  |55   30  12   4  1|   |14   5   2  1  1| | 0  5   2  1  1 |  | 0  0  2  1  1 |
  |...               |   |...             | |...             |  |...            |
where T denotes transposition and [1, 1, 2, 5, 14,...] is the sequence of Catalan numbers A000108. (End)

Peter Bala, Factorisations of some Riordan arrays as infinite products
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 27, 29.
Paul Barry, d-orthogonal polynomials, Fuss-Catalan matrices and lattice paths, arXiv:2505.16718 [math.CO], 2025. See p. 21.
Naiomi Cameron and J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.
Sheng-Liang Yang and L. J. Wang, Taylor expansions for the m-Catalan numbers, Australasian Journal of Combinatorics, Volume 64(3) (2016), Pages 420-431.

Successive columns: A001764, A006013, A001764, A006629, A102893, A006630, A102594, A006631; row sums: A098746; see also A092276.

Table[(k + 1) Binomial[3 n - 2 k, 2 n - k]/(2 n - k + 1), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jun 28 2017 *)

T(n,k):=((k+1)*binomial(3*n-2*k,2*n-k))/(2*n-k+1); /* Vladimir Kruchinin, Nov 01 2011 */

T(n, k) = Sum_{j>=0} T(n-1, k-1+j)*A000108(j); T(0, 0) = 1; T(n, k) = 0 if k < 0 or if k > n.
G.f.: 1/(1 - x*y*TernaryGF) = 1 + (y)x + (y+y^2)x^2 + (3y+2y^2+y^3)x^3 +... where TernaryGF = 1 + x + 3x^2 + 12x^3 + ... is the GF for A001764. - David Callan, Aug 27 2009
T(n, k) = ((k+1)*binomial(3*n-2*k,2*n-k))/(2*n-k+1). - Vladimir Kruchinin, Nov 01 2011

A122649 Difference between the double factorial of the n-th nonnegative odd number and the double factorial of the n-th nonnegative even number.

Original entry on oeis.org

0, 1, 7, 57, 561, 6555, 89055, 1381905, 24137505, 468934515, 10033419375, 234484536825, 5943863027025, 162446292283275, 4761954230608575, 149048910271886625, 4961463912662882625, 175022432901300859875

Offset: 1

Peter C. Heinig (algorithms(AT)gmx.de), Sep 21 2006

			a(1) = 0, since 1!! - 0!! = 1 - 1 = 0, where the usual convention 0!! = 1 has been heeded. Note that 1 is the first nonnegative odd and 0 the first nonnegative even number.
a(4) = 57, since 7!! - 6!! = 1*3*5*7 - 6*4*2*1 = 105 - 48 = 57.

Vincenzo Librandi, Table of n, a(n) for n = 1..410

Cf. A006882, A000165, A006013.

for n from 1 to 24 do: l[n]:=product(2*k-1, k=1..n); od: r[1]:=1; for n from 2 to 24 do: r[n]:=product(2*k, k=1..n-1); od; for k from 1 to 24 do: a[k]:=l[k]-r[k]; od;

#[[2]]-#[[1]]&/@Partition[Range[0,40]!!,2] (* Harvey P. Dale, Feb 19 2013 *)
Rest[Range[0, 100]! CoefficientList[Series[-1 + 1/Sqrt[1 - 2 x] + Log[1 - 2 x]/2, {x, 0, 800}], x]] (* Vincenzo Librandi, Jun 24 2016 *)

a(n) = (2*n - 1)!! - (2*n - 2)!! = A006882(2*n - 1) - A000165(n - 1).
From Peter Bala, Jun 22 2016: (Start)
a(1) = 0, a(2) = 1 and for n >= 3, a(n) = (4*n - 5)*a(n-1) - (2*n - 4)*(2*n - 3)*a(n-2).
E.g.f. assuming an offset of 0: A(x) = 1/(1 - 2*x)^(3/2) - 1/(1 - 2*x) = x + 7*x^2/2! + 57*x^3/3! + ....
A( Sum_{n >= 1} n^(n-2)*x^n/n! ) = Sum_{n >= 1} n^(n+1)*x^n/n!.
Series reversion (A(x)) = 1/2*Sum_{n >= 1} (-1)^(n+1)*1/(n+1)*
binomial(3*n + 1,n)*x^n. Cf. A006013.(End)
E.g.f.: -1 + 1/sqrt(1-2*x) + log(1-2*x)/2. - Ilya Gutkovskiy, Jun 23 2016

A152555 Coefficients in a q-analog of the function LambertW(-2x)/(-2x), as a triangle read by rows.

Original entry on oeis.org

1, 2, 7, 5, 30, 42, 42, 14, 143, 297, 462, 495, 363, 198, 42, 728, 2002, 4004, 6006, 7436, 7436, 6292, 4290, 2288, 858, 132, 3876, 13260, 31824, 58604, 91364, 122876, 145535, 153361, 143936, 120185, 87971, 56329, 29939, 12584, 3575, 429, 21318, 87210

Offset: 0

Paul D. Hanna, Dec 07 2008

			Triangle begins:
  1;
  2;
  7,5;
  30,42,42,14;
  143,297,462,495,363,198,42;
  728,2002,4004,6006,7436,7436,6292,4290,2288,858,132;
  3876,13260,31824,58604,91364,122876,145535,153361,143936,120185,87971,56329,29939,12584,3575,429;
  21318,87210,242250,519384,945744,1508070,2165664,2826420,3392520,3756626,3853322,3662106,3221330,2613240,1944324,1313760,794614,420784,185640,64090,14586,1430;...
where row sums = 2*(2*n+2)^(n-1) (A097629).
Row sums at q=-1 = 2*(2*n+2)^[(n-1)/2] (A152556).
The generating function starts:
A(x,q) = 1 + 2*x + (7 + 5*q)*x^2/faq(2,q) + (30 + 42*q + 42*q^2 + 14*q^3)*x^3/faq(3,q) + (143 + 297*q + 462*q^2 + 495*q^3 + 363*q^4 + 198*q^5 + 42*q^6)*x^4/faq(4,q) + ...
The q-factorial of n is faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1): faq(0,q)=1, faq(1,q)=1, faq(2,q)=(1+q), faq(3,q)=(1+q)*(1+q+q^2), faq(4,q)=(1+q)*(1+q+q^2)*(1+q+q^2+q^3), ...
Special cases.
q=0: A(x,0) = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 +... (A006013)
q=1: A(x,1) = 1 + 2*x + 12/2*x^2 + 128/6*x^3 + 2000/24*x^4 + 41472/120*x^5 +...
q=2: A(x,2) = 1 + 2*x + 17/3*x^2 + 394/21*x^3 + 21377/315*x^4 + 2537724/9765*x^5 +...
q=3: A(x,3) = 1 + 2*x + 22/4*x^2 + 912/52*x^3 + 126692/2080*x^4 + 56277344/251680*x^5 +...

Paul D. Hanna, Rows 0 to 30 of the triangle, flattened.
Eric Weisstein's World of Mathematics, q-Exponential Function.
Eric Weisstein's World of Mathematics, q-Factorial.

Cf. A097629 (row sums), A006013 (column 0), A000108 (right border), A152559.
Cf. A152556 (q=-1), A152557 (q=2), A152558 (q=3).
Cf. variants: A152290, A152550.

{T(n,k)=local(e_q=1+sum(j=1,n,x^j/prod(i=1,j,(q^i-1)/(q-1))), LW2_q=serreverse(x/(e_q+x*O(x^n))^2)/x); polcoeff(polcoeff(LW2_q+x*O(x^n),n,x)*prod(i=1,n,(q^i-1)/(q-1))+q*O(q^k),k,q)}

G.f.: A(x,q) = Sum_{n>=0} Sum_{k=0..n*(n-1)/2} T(n,k)*q^k*x^n/faq(n,q), where faq(n,q) is the q-factorial of n.
G.f.: A(x,q) = (1/x)*Series_Reversion( x/e_q(x,q)^2 ) where e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) is the q-exponential function.
G.f. satisfies: A(x,q) = e_q( x*A(x,q), q)^2 and A( x/e_q(x,q)^2, q) = e_q(x,q)^2.
G.f. at q=1: A(x,1) = LambertW(-2*x)/(-2*x).
Row sums at q=+1: Sum_{k=0..n*(n-1)/2} T(n,k) = 2*(2*n+2)^(n-1).
Row sums at q=-1: Sum_{k=0..n*(n-1)/2} T(n,k)*(-1)^k = 2*(2*n+2)^[(n-1)/2].
Sum_{k=0..n*(n-1)/2} T(n,k)*exp(2*Pi*I*k/n) = 2 for n>=1; i.e., the n-th row sum at q = exp(2*Pi*I/n), the n-th root of unity, equals 2 for n>=1. - Vladeta Jovovic
Sum_{k=0..binomial(n,2)} T(n,k)*q^k = Sum_{pi} 2*(2*n+1)!/(2*n-k+2)!*faq(n,q)/Product_{i=1..n} e(i)!*faq(i,q)^e(i), where pi runs over all nonnegative integer solutions to e(1)+2*e(2)+...+n*e(n) = n and k = e(1)+e(2)+...+e(n). - Vladeta Jovovic, Dec 07 2008

A167763 Pendular triangle (p=0), read by rows, where row n is formed from row n-1 by the recurrence: if n > 2k, T(n,k) = T(n,n-k) + T(n-1,k), otherwise T(n,k) = T(n,n-1-k) + p*T(n-1,k), for n >= k <= 0, with T(n,0) = 1 and T(n,n) = 0^n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 7, 4, 1, 0, 1, 5, 12, 12, 5, 1, 0, 1, 6, 18, 30, 18, 6, 1, 0, 1, 7, 25, 55, 55, 25, 7, 1, 0, 1, 8, 33, 88, 143, 88, 33, 8, 1, 0, 1, 9, 42, 130, 273, 273, 130, 42, 9, 1, 0, 1, 10, 52, 182, 455, 728, 455, 182, 52, 10, 1, 0

Offset: 0

Philippe Deléham, Nov 11 2009

See A118340 for definition of pendular triangles and pendular sums.

The last five rows in the example section appear in the table on p. 8 of Getzler. Cf. also A173075. - Tom Copeland, Jan 22 2020

			Triangle begins:
  1;
  1,  0;
  1,  1,  0;
  1,  2,  1,  0;
  1,  3,  3,  1,  0;
  1,  4,  7,  4,  1,  0;
  1,  5, 12, 12,  5,  1,  0; ...

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
E. Getzler, The semi-classical approximation for modular operads, arXiv:alg-geom/9612005, 1996.

Cf. this sequence (p=0), A118340 (p=1), A118345 (p=2), A118350 (p=3).

Cf. A001764, A006013, A006629, A093951, A102893, A173075.

function T(n,k,p)
  if k lt 0 or n lt k then return 0;
  elif k eq 0 then return 1;
  elif k eq n then return 0;
  elif n gt 2*k then return T(n,n-k,p) + T(n-1,k,p);
  else return T(n,n-k-1,p) + p*T(n-1,k,p);
  end if;
  return T;
end function;
[T(n,k,0): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2021

T[n_, k_, p_]:= T[n,k,p] = If[nG. C. Greubel, Feb 17 2021 *)

{T(n,k)=if(k==0,1,if(n==k,0,if(n>2*k,binomial(n+k+1,k)*(n-2*k+1)/(n+k+1),T(n,n-1-k))))} \\ Paul D. Hanna, Nov 12 2009

@CachedFunction
def T(n, k, p):
    if (k<0 or n2*k): return T(n,n-k,p) + T(n-1,k,p)
    else: return T(n, n-k-1, p) + p*T(n-1, k, p)
flatten([[T(n, k, 0) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 17 2021

T(2n+m) = [A001764^(m+1)](n), i.e., the m-th lower semi-diagonal forms the self-convolution (m+1)-power of A001764.
If n > 2k, T(n,k) = binomial(n+k+1,k)*(n-2k+1)/(n+k+1), else T(n,k) = T(n,n-1-k), with conditions: T(n,0)=1 for n>=0 and T(n,n)=0 for n > 0. - Paul D. Hanna, Nov 12 2009
Sum_{k=0..n} T(n, k, p=0) = A093951(n). - G. C. Greubel, Feb 17 2021

cp's OEIS Frontend

A102893 Number of noncrossing trees with n edges and having degree of the root at least 2.

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A143363 Number of ordered trees with n edges and having no protected vertices. A protected vertex in an ordered tree is a vertex at least 2 edges away from its leaf descendants.

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A063030 Reversion of y - y^2 - y^4 + y^5.

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A278880 Triangle where g.f. S = S(x,m) satisfies: S = x/(G(-S^2)*G(-m*S^2)) such that G(x) = 1 + x*G(x)^2 is the g.f. of the Catalan numbers (A000108), as read by rows of coefficients T(n,k) of x^(2*n-1)*m^k in S(x,m) for n>=1, k=0..n-1.

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A286784 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

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A085405 Common residues of binomial(3n+2,n+1)/(3n+2) modulo 2.

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A110616 A convolution triangle of numbers based on A001764.

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A278880 Triangle where g.f. S = S(x,m) satisfies: S = x/(G(-S^2)G(-mS^2)) such that G(x) = 1 + xG(x)^2 is the g.f. of the Catalan numbers (A000108), as read by rows of coefficients T(n,k) of x^(2n-1)*m^k in S(x,m) for n>=1, k=0..n-1.

A152555 Coefficients in a q-analog of the function LambertW(-2x)/(-2x), as a triangle read by rows.