A075045 -id:A075045 - cp's OEIS frontend

A062236 Sum of the levels of all nodes in all noncrossing trees with n edges.

1, 8, 58, 408, 2831, 19496, 133638, 913200, 6226591, 42387168, 288194424, 1957583712, 13286865060, 90126841064, 611029568078, 4140789069408, 28050809681679, 189964288098632, 1286119453570746, 8705397371980728, 58912358137385559, 398607288093924192, 2696583955707785256

Offset: 1

Emeric Deutsch, Jun 30 2001

nonn

Harry J. Smith, Table of n, a(n) for n=1..200
Emeric Deutsch and M. Noy, New statistics on non-crossing trees, in: Formal Power Series and Algebraic Combinatorics (Proceedings of the 12th International Conference, FPSAC'00, Moscow, Russia, 2000), pp. 667-676, Springer, Berlin, 2000.
Emeric Deutsch and M. Noy, Statistics on non-crossing trees, Discrete Math., 254 (2002), 75-87 (see Th. 6). [From _N. J. A. Sloane_, Dec 17 2012]
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.

Cf. A001764, A036829, A358091.

Cf. A006256, A036829, A075045, A386617.

a := n -> add(2^(n-2-i)*(n-i)*(3*n-3*i-1)*binomial(3*n,i),i=0..n-1)/n;
A062236 := n -> 2^(n-2)*(3*n-1)*hypergeom([-3*n,1-n,-n+4/3], [-n,-n+1/3], -1/2):
seq(simplify(A062236(n)), n = 1..29); # Peter Luschny, Oct 28 2022

Table[Sum[2^(n-2-k)*(n-k)*(3*n-3*k-1)*Binomial[3*n,k],{k,0,n-1}]/n,{n,1,20}] (* Vaclav Kotesovec, Oct 13 2012 *)

{ for (n=1, 200, a=sum(i=0, n-1, 2^(n-2-i)*(n-i)*(3*n-3*i-1)*binomial(3*n, i))/n; write("b062236.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 03 2009

G.f.: g*(g-1)/(3-2*g)^2, where function g=g(x) satisfies g=1+xg^3, and can be expressed as g(x) = 2*sin(arcsin(3*sqrt(3*x)/2)/3)/sqrt(3*x). [Corrected by Max Alekseyev, Oct 27 2022]
g(x) = Sum_{n >= 0} binomial(3*n,n) / (2*n+1) * x^n. - Max Alekseyev, Oct 27 2022
Recurrence: 8*n*(2*n-1)*a(n) = 6*(36*n^2-45*n+10)*a(n-1) - 81*(3*n-5)*(3*n-1)*a(n-2). - Vaclav Kotesovec, Oct 13 2012
a(n) ~ 3^(3*n)/2^(2*n+2). - Vaclav Kotesovec, Oct 13 2012
a(n) = Sum_{i=0..n-1} C(3*i-1,i)*C(3*(n-i),n-i-1). - Vladimir Kruchinin, Jun 09 2020
a(n) = 2^(n-2)*(3*n-1)*hypergeometric([-3*n, 1-n, -n+4/3], [-n, -n+1/3], -1/2). The a(n) are values of the polynomials A358091. - Peter Luschny, Oct 28 2022
From Seiichi Manyama, Jul 26 2025: (Start)
G.f.: g/(1-3*g)^2 where g*(1-g)^2 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/2) * log( Sum_{k>=0} binomial(3*k-1,k)*x^k ). (End)
From Seiichi Manyama, Jul 29 2025: (Start)
a(n) = Sum_{k=0..n-1} binomial(3*k-1+l,k) * binomial(3*n-3*k-l,n-k-1) for every real number l. This is a generalization of a formula by Vladimir Kruchinin, Jun 09 2020.
a(n) = Sum_{k=0..n-1} 2^(n-k-1) * binomial(3*n,k).
a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(2*n+k,k). (End)

A049235 Sum of balls on the lawn for the s=3 tennis ball problem.

Original entry on oeis.org

0, 6, 75, 708, 5991, 47868, 369315, 2783448, 20631126, 151026498, 1094965524, 7878119760, 56330252412, 400703095284, 2838060684483, 20027058300144, 140874026880204, 988194254587242, 6915098239841331, 48285969880645908, 336521149274459979

Offset: 0

N. J. A. Sloane, Jan 19 2003

nonn

Robert Israel, Table of n, a(n) for n = 0..1202
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (S_n for s=3).

The four sequences T_n, Y_n, A_n, S_n for s=2 are A000108, A000302, A000346, A031970, for s=3, A001764, A006256, A075045, this sequence, for s=4, A002293, A078995, A078999, A078516.

Cf. A079486.

T := (n,s)->binomial(s*n,n)/((s-1)*n+1); Y := (n,s)->add(binomial(s*k,k)*binomial(s*(n-k),n-k),k=0..n); A := (n,s)->Y(n+1,s)/2-(1/2)*((2*s-3)*n+2*s-2)*T(n+1,s); S := (n,s)->(1/2)*(s*n^2+(3*s-1)*n+2*s)*T(n+1,s)-Y(n+1,s)/2;
F := 3*(2-3*t)*t*((t-1)*(3*t-1))^(-3);  G := t*(t-1)^2;   Ginv := RootOf(G-x,t);
ogf := series(eval(F,t=Ginv), x=0, 20);

a[n_] := a[n] = Switch[n, 0, 0, 1, 6, 2, 75, 3, 708, 4, 5991, _, -((1/(8*(2*(n-5)^2 + 25*(n-5) + 78)))*(-(531441*(n-5)^2* a[n-5]) + 196830*(n-5)^2*a[n-4] - 24057*(n-5)^2*a[n-3] + 1809*(n-5)^2*a[n-2] - 232*(n-5)^2*a[n-1] - 1594323*(n-5)*a[n-5] + 747954*(n-5)*a[n-4] - 120285*(n-5)*a[n-3] + 16362*(n-5)*a[n-2] - 2798*(n-5)*a[n-1] - 1180980*a[n-5] + 656100*a[n-4] - 131220*a[n-3] + 36825*a[n-2] - 8352*a[n-1]))];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 02 2023, after Robert Israel *)

a(n) is asymptotic to c*sqrt(n)*(27/4)^n with c=2.4... - Benoit Cloitre, Jan 26 2003, c = 81*sqrt(3/Pi)/32 = 2.4735502165085321... - Vaclav Kotesovec, Feb 07 2019
G.f.: F(G^(-1)(x)) where F = 3*(2-3*t)*t*((t-1)*(3*t-1))^(-3) and G = t*(t-1)^2.  - Mark van Hoeij, Oct 30 2011
D-finite with recurrence (531441*n^2 + 1594323*n + 1180980)*a(n) + (-196830*n^2 - 747954*n - 656100)*a(n + 1) + (24057*n^2 + 120285*n + 131220)*a(n + 2) + (-1809*n^2 - 16362*n - 36825)*a(n + 3) + (232*n^2 + 2798*n + 8352)*a(n + 4) + (-16*n^2 - 200*n - 624)*a(n + 5) = 0. - Robert Israel, Jun 20 2019

A386617 a(n) = Sum_{k=0..n-1} binomial(3k+1,k) binomial(3n-3k,n-k-1).

Original entry on oeis.org

0, 1, 10, 81, 610, 4436, 31626, 222681, 1554772, 10790721, 74560728, 513452604, 3526463304, 24168921568, 165357919850, 1129724254953, 7709039995368, 52551835079699, 357930487932282, 2436038623348521, 16568626556643738, 112626521811112464, 765201654587796312, 5196570956399432796

Offset: 0

Seiichi Manyama, Jul 27 2025

nonn

Cf. A006256, A036829, A062236, A075045.

Cf. A006419, A386612, A386614, A386616.

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

G.f.: g^3 * (g-1)/(3-2*g)^2 where g=1+x*g^3.
G.f.: g/((1-g)^2 * (1-3*g)^2) where g*(1-g)^2 = x.
a(n) = Sum_{k=0..n-1} binomial(3*k+1+l,k) * binomial(3*n-3*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 2^(n-k-1) * binomial(3*n+2,k).
a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(2*n+k+2,k).

A361960 Total semiperimeter of 2-Fuss-Catalan polyominoes of length 2n.

Original entry on oeis.org

2, 12, 71, 430, 2652, 16576, 104652, 665874, 4263050, 27430260, 177233355, 1149159336, 7473264736, 48725661120, 318403991656, 2084753927898, 13673789668854, 89825336129620, 590901795716925, 3892055708986830, 25664871706721940, 169414775012098560, 1119378775384200240, 7402571891557073400, 48993463632294517752, 324501821324483687856

Offset: 1

R. J. Mathar, Mar 31 2023

Toufik Mansour, I. L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457, Table 2

Cf. A024482 (1-Fuss-Catalan), A075045 (total area), A361961 (3-Fuss-Catalan).

Per := proc(s,p,n)
    local i,j,a ;
    a := 0 ;
    for i from 0 to n-1 do
    for j from 0 to n-1-i do
        a := a+ (-1)^j*p^(n+1+i+(s+1)*j) *binomial(n-1+i,i)*binomial(n,j)*binomial(n+s*j,n-1-i-j)/(1-p)^(i+j) ;
    end do:
    end do:
    expand(a/n) ;
    factor(%) ;
end proc:
Per1std := proc(s,n)
    local p;
    Per(s,p,n) ;
    diff(%,p) ;
    factor(%) ;
    subs(p=1,%) ;
end proc:
seq(Per1std(2,n),n=1..30) ;

Conjecture: D-finite with recurrence 4*n*(2*n+1)*a(n) -6*n*(11*n-5)*a(n-1) +3*(43*n^2-169*n+130)*a(n-2) -36*(3*n-8)*(3*n-10)*a(n-3)=0.

cp's OEIS Frontend

A062236 Sum of the levels of all nodes in all noncrossing trees with n edges.

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A049235 Sum of balls on the lawn for the s=3 tennis ball problem.

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Maple

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A386617 a(n) = Sum_{k=0..n-1} binomial(3k+1,k) binomial(3n-3k,n-k-1).

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PARI

Formula

A361960 Total semiperimeter of 2-Fuss-Catalan polyominoes of length 2n.

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Maple

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cp's OEIS Frontend

A062236 Sum of the levels of all nodes in all noncrossing trees with n edges.

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Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A049235 Sum of balls on the lawn for the s=3 tennis ball problem.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A386617 a(n) = Sum_{k=0..n-1} binomial(3*k+1,k) * binomial(3*n-3*k,n-k-1).

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Author

Keywords

Crossrefs

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PARI

Formula

A361960 Total semiperimeter of 2-Fuss-Catalan polyominoes of length 2n.

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Author

Keywords

Links

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Programs

Maple

Formula

A386617 a(n) = Sum_{k=0..n-1} binomial(3k+1,k) binomial(3n-3k,n-k-1).