A386617 -id:A386617 - cp's OEIS frontend

A006419 a(n) = 2^(2n+1) - C(2n+3,n+1) + C(2*n+1,n).

0, 1, 7, 37, 176, 794, 3473, 14893, 63004, 263950, 1097790, 4540386, 18696432, 76717268, 313889477, 1281220733, 5219170052, 21224674118, 86188320962, 349550141078, 1416102710912, 5731427140268, 23177285611082, 93655986978002, 378195990166136, 1526289367335244

Offset: 0

N. J. A. Sloane

nonn

Number of rooted isthmusless planar maps with n+1 faces and 2 vertices. - Dan Drake, Aug 08 2005
a(n) = total area below all Dyck (n+1)-paths and above the lowest possible Dyck path, namely, UDUD...UD (taking upsteps of unit length). For example, the areas below the 5 Dyck 3-paths UUUDDD, UUDUDD, UDUUDD, UUDDUD, UDUDUD are 3,2,1,1,0 respectively, yielding a(2)=3+2+1+1+0=7. - David Callan, Jul 03 2006
Convolution of A000245 and A000302 (powers of 4).- Philippe Deléham, Jun 02 2013

			G.f. = x + 7*x^2 + 37*x^3 + 176*x^4 + 794*x^5 + 3473*x^6 + 14893*x^7 + 63004*x^8 + ...

D. Phulara and L. W. Shapiro, Descendants in ordered trees with a marked vertex, Congressus Numerantium, 205 (2011), 121-128.
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..1660
Jason Bandlow and Kendra Killpatrick, An area-to-inv bijection between Dyck paths and 312-avoiding permutations,Electron. J. Combin. 8 (2001), no. 1, Research Paper 40, 16 pp.
Miklós Bóna, Surprising Symmetries in Objects Counted by Catalan Numbers, Electronic J. Combin., 19 (2012), #P62, eq. (5).
Pudwell, Lara; Scholten, Connor; Schrock, Tyler; Serrato, Alexa Noncontiguous pattern containment in binary trees, ISRN Comb. 2014, Article ID 316535, 8 p. (2014), Table 1.
R. P. Stanley, F. Zanello, The Catalan case of Armstrong's conjecture on core partitions, arXiv preprint arXiv:1312.4352 [math.CO], 2013.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259.

A diagonal of A342981.

Cf. A386612, A386614, A386616, A386617.

f := n->2^(2*n+1)-binomial(2*n+3,n+1)+binomial(2*n+1,n); seq(f(n), n=0..30);

Table[2^(2 n + 1) - Binomial[2 n + 3, n + 1] +
Binomial[2 n + 1, n], {n, 0, 30}] (* Wesley Ivan Hurt, Mar 30 2014 *)

a(n):=sum(binomial(2*(n+1),n-k-1),k,0,n); /* Vladimir Kruchinin, Oct 23 2016 */

a(n+1) = Sum_{k=0..n} (n-k)*A000108(n-k)*A001700(k). - Philippe Deléham, Jan 25 2004
G.f.: c(x)^3*x/(1-4x) where c(x) = g.f. for the Catalan numbers A000108. - Philippe Deléham, Jun 02 2013
a(n) = Integral_{x=0..4} x^n*W(x)*dx, n >= 0, is the integral representation as n-th moment of a signed weight function W(x), where W(x) = W_a(x) + W_c(x), with W_a(x) = 2*Dirac(x-4), which is the discrete (atomic) part, and W_c(x) = (1/(2*Pi))*(1-x)*sqrt(x/(4-x)) is the continuous part of W(x): W_c(0) = W_c(1) = 0, W_c(x) > 0 for x < 1, lim_{x->4} W_c(x) = -oo. - Karol A. Penson, Jul 31 2013 [edited by Michel Marcus, Mar 14 2020]
(n+2)*a(n) + (-9*n-10)*a(n-1) + 2*(12*n+1)*a(n-2) + 8*(-2*n+3)*a(n-3) = 0. - R. J. Mathar, Mar 30 2014
a(n) = Sum_{k=0..n} binomial(2*(n+1), n-k-1). - Vladimir Kruchinin, Oct 23 2016
0 = a(n)*(+256*a(n+1) - 992*a(n+2) + 520*a(n+3) - 72*a(n+4)) + a(n+1)*(+224*a(n+1) + 344*a(n+2) - 398*a(n+3) + 70*a(n+4)) + a(n+2)*(+6*a(n+2) + 59*a(n+3) - 17*a(n+4)) + a(n+3)*(-a(n+3) + a(n+4)), for all n >= 0. - Michael Somos, Oct 23 2016
a(n) = [x^n] x/((1 - 2*x)*(1 - x)^(n+3)). - Ilya Gutkovskiy, Oct 25 2017
From Seiichi Manyama, Jul 29 2025: (Start)
a(n) = Sum_{k=0..n-1} binomial(2*k+1+l,k) * binomial(2*n-2*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 2^(n-k-1) * binomial(n+k+2,k). (End)

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

Original entry on oeis.org

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)

A386612 a(n) = Sum_{k=0..n-1} binomial(4k+1,k) binomial(4n-4k,n-k-1).

Original entry on oeis.org

0, 1, 13, 142, 1464, 14689, 145154, 1420812, 13818784, 133793940, 1291073809, 12426782294, 119371355672, 1144851458526, 10965655515588, 104919037771224, 1002960800712720, 9580390527192940, 91453374122574372, 872513477065735768, 8320168165323802464, 79305962393873976417

Offset: 0

Seiichi Manyama, Jul 27 2025

nonn

Cf. A078995, A308523, A386565, A386611.

Cf. A006419, A386614, A386616, A386617.

a(n) = sum(k=0, n-1, binomial(4*k+1, k)*binomial(4*n-4*k, n-k-1));

G.f.: g^3 * (g-1)/(4-3*g)^2 where g=1+x*g^4.
G.f.: g/((1-g)^2 * (1-4*g)^2) where g*(1-g)^3 = x.
a(n) = Sum_{k=0..n-1} binomial(4*k+1+l,k) * binomial(4*n-4*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n+2,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k+2,k).
D-finite with recurrence 13122*n*(3*n+2)*(3*n+1)*a(n) +81*(-124803*n^3+284553*n^2-210740*n+42140)*a(n-1) +24*(7476768*n^3-29253744*n^2+37920106*n-16562575)*a(n-2) +40960*(-26344*n^3+148032*n^2-282329*n+185874)*a(n-3) +55050240*(2*n-5)*(4*n-13)*(4*n-11)*a(n-4)=0. - R. J. Mathar, Aug 10 2025

A386614 a(n) = Sum_{k=0..n-1} binomial(5k+1,k) binomial(5n-5k,n-k-1).

Original entry on oeis.org

0, 1, 16, 220, 2880, 36850, 465536, 5834852, 72744640, 903525715, 11191199200, 138323478980, 1706860996096, 21034268215120, 258934785258240, 3184696786012500, 39140208951032960, 480734044749851305, 5901368553964031600, 72410017973538837880, 888114187330722044800, 10888921795007470528060

Offset: 0

Seiichi Manyama, Jul 27 2025

nonn

Cf. A079678, A386367, A386566, A386613.

Cf. A006419, A386612, A386616, A386617.

a(n) = sum(k=0, n-1, binomial(5*k+1, k)*binomial(5*n-5*k, n-k-1));

G.f.: g^3 * (g-1)/(5-4*g)^2 where g=1+x*g^5.
G.f.: g/((1-g)^2 * (1-5*g)^2) where g*(1-g)^4 = x.
a(n) = Sum_{k=0..n-1} binomial(5*k+1+l,k) * binomial(5*n-5*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(5*n+2,k).
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(4*n+k+2,k).
D-finite with recurrence +35651584*n*(4*n+1)*(2*n+1)*(4*n-1)*a(n) +8192*(56348704*n^4-268019168*n^3+418502324*n^2-264019618*n+57303885)*a(n-1) +160*(-65524820000*n^4+314102050000*n^3-463341186250*n^2+159732814775*n+76118151939)*a(n-2) +62500*(660806875*n^4-1813661250*n^3-5080986250*n^2+20705993100*n-17279228304)*a(n-3) +308935546875*(5*n-11)*(5*n-14)*(5*n-13)*(5*n-17)*a(n-4)=0. - R. J. Mathar, Aug 10 2025

A386616 a(n) = Sum_{k=0..n-1} binomial(6k+1,k) binomial(6n-6k,n-k-1).

Original entry on oeis.org

0, 1, 19, 315, 5000, 77785, 1196667, 18282742, 278031900, 4214278350, 63723788295, 961789682008, 14495501585664, 218216042892175, 3281961694927950, 49322417450239980, 740753733463215604, 11118981305235476010, 166821561372208253850, 2501861335268901337425, 37507747177968865536840

Offset: 0

Seiichi Manyama, Jul 27 2025

nonn

Cf. A079679, A386368, A386567, A386615.

Cf. A006419, A386612, A386614, A386617.

a(n) = sum(k=0, n-1, binomial(6*k+1, k)*binomial(6*n-6*k, n-k-1));

G.f.: g^3 * (g-1)/(6-5*g)^2 where g=1+x*g^6.
G.f.: g/((1-g)^2 * (1-6*g)^2) where g*(1-g)^5 = x.
a(n) = Sum_{k=0..n-1} binomial(6*k+1+l,k) * binomial(6*n-6*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(6*n+2,k).
a(n) = Sum_{k=0..n-1} 6^(n-k-1) * binomial(5*n+k+2,k).

A075045 Coefficients A_n for the s=3 tennis ball problem.

Original entry on oeis.org

1, 9, 69, 502, 3564, 24960, 173325, 1196748, 8229849, 56427177, 386011116, 2635972920, 17974898872, 122430895956, 833108684637, 5664553564440, 38488954887171, 261369752763963, 1774016418598269, 12035694958994142, 81624256468292016, 553377268856455968

Offset: 0

N. J. A. Sloane, Jan 19 2003

nonn

T. Amdeberhan, Integrality of a sum.
Roland Bacher, On generating series of complementary plane trees, arXiv:math/0409050 [math.CO], 2004.
Toufik Mansour and I. L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457, table 1.
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), pp. 307-344 (A_n for s=3).

See A049235 for more information.

Cf. A006256, A036829, A062236, A386617.

FussArea := proc(s,n)
    local a,i,j ;
    a := binomial((s+1)*n,n)*n/(s*n+1) ; ;
    add(j *(n-j) *binomial((s+1)*j,j) *binomial((s+1)*(n-j),n-j) /(s*j+1) /(s*(n-j)+1),j=0..n) ;
    a := a+binomial(s+1,2)*% ;
    for j from 0 to n-1 do
        for i from 0 to j do
            i*(j-i) /(s*i+1) /(s*(j-i)+1) /(n-j)
            *binomial((s+1)*i,i) *binomial((s+1)*(j-i),j-i)
            *binomial((s+1)*(n-j)-2,n-1-j) ;
            a := a-%*binomial(s+1,2) ;
        end do:
    end do:
    a ;
end proc:
seq(FussArea(2,n),n=1..30) ; # R. J. Mathar, Mar 31 2023

FussArea[s_, n_] := Module[{a, i, j, pc}, a = Binomial[(s + 1)*n, n]*n/(s*n + 1); pc = Sum[j*(n - j)*Binomial[(s + 1)*j, j]*Binomial[(s + 1)*(n - j), n - j]/(s*j + 1)/(s*(n - j) + 1), {j, 0, n}]; a = a + Binomial[s + 1, 2]*pc; For[j = 0, j <= n - 1 , j++, For[i = 0, i <= j, i++, pc = i*(j - i)/(s*i + 1)/(s*(j - i) + 1)/(n - j)*Binomial[(s + 1)*i, i]* Binomial[(s + 1)*(j - i), j - i]*Binomial[(s + 1)*(n - j) - 2, n - 1 - j]; a = a - pc*Binomial[s + 1, 2]; ]]; a];
Table[FussArea[2, n], {n, 1, 30}] (* Jean-François Alcover, Apr 02 2023, after R. J. Mathar *)

G.f.: seems to be (3*g-1)^(-2)*(1-g)^(-3) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011
Conjecture: D-finite with recurrence 8*(2*n+3)*(7*n+1)*(n+1)*a(n) +6*(-252*n^3-477*n^2-220*n-11)*a(n-1) +81*(7*n+8)*(3*n-1)*(3*n+1)*a(n-2)=0. - Jean-François Alcover, Feb 07 2019
a(n) = (3n+2)*(n+1)*binomial(3n+3,n+1)/2/(2n+3) - A049235(n). [Merlini Theorem 2.5 for s=3] - R. J. Mathar, Oct 01 2021
From Seiichi Manyama, Jul 28 2025: (Start)
a(n) = Sum_{k=0..n} binomial(3*k+3+l,k) * binomial(3*n-3*k-l,n-k) for every real number l.
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+4,k).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(2*n+k+3,k). (End)

A385004 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n,k).

Original entry on oeis.org

1, 5, 31, 200, 1311, 8665, 57556, 383556, 2561871, 17140007, 114819351, 769925568, 5166845124, 34696155564, 233113911208, 1566926561740, 10536427052463, 70872688450083, 476854924775869, 3209222876463192, 21602639249766951, 145444151677134153, 979397744169608784

Offset: 0

Seiichi Manyama, Jul 30 2025

Cf. A000244, A100192, A385498, A386699.
Cf. A005809, A066380, A165817, A371813, A386700.
Cf. A001764, A386617.

Table[(27/4)^n - Binomial[3*n, n] * (-1 + Hypergeometric2F1[1, -2*n, 1 + n, -1/2]), {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)

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

a(n) = [x^n] 1/((1-3*x) * (1-x)^(2*n)).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(2*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 8*n*(2*n - 1)*(15*n - 23)*a(n) = 6*(540*n^3 - 1503*n^2 + 1239*n - 320)*a(n-1) - 81*(3*n - 5)*(3*n - 4)*(15*n - 8)*a(n-2).
a(n) ~ 3^(3*n) / 2^(2*n+1) * (1 + 5/(3*sqrt(3*Pi*n))). (End)
G.f.: g/(3-2*g)^2 where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^2*(9-4*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 17 2025

cp's OEIS Frontend

A006419 a(n) = 2^(2*n+1) - C(2*n+3,n+1) + C(2*n+1,n).

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

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A386612 a(n) = Sum_{k=0..n-1} binomial(4*k+1,k) * binomial(4*n-4*k,n-k-1).

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A386614 a(n) = Sum_{k=0..n-1} binomial(5*k+1,k) * binomial(5*n-5*k,n-k-1).

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A386616 a(n) = Sum_{k=0..n-1} binomial(6*k+1,k) * binomial(6*n-6*k,n-k-1).

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A075045 Coefficients A_n for the s=3 tennis ball problem.

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A385004 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n,k).

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A006419 a(n) = 2^(2n+1) - C(2n+3,n+1) + C(2*n+1,n).

A386612 a(n) = Sum_{k=0..n-1} binomial(4k+1,k) binomial(4n-4k,n-k-1).

A386614 a(n) = Sum_{k=0..n-1} binomial(5k+1,k) binomial(5n-5k,n-k-1).

A386616 a(n) = Sum_{k=0..n-1} binomial(6k+1,k) binomial(6n-6k,n-k-1).