A386612 -id:A386612 - 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)

A308523 Number of essentially simple rooted toroidal triangulations with n vertices.

Original entry on oeis.org

0, 1, 10, 97, 932, 8916, 85090, 810846, 7719048, 73431340, 698187400, 6635738209, 63047912372, 598885073788, 5687581936284, 54005562798252, 512728901004816, 4867263839614716, 46199494669833400, 438481077306427924, 4161316466910824272

Offset: 0

Nicolas Bonichon, Jun 05 2019

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..499
Nicolas Bonichon, Éric Fusy, Benjamin Lévêque, A bijection for essentially 3-connected toroidal maps, arXiv:1907.04016 [math.CO], 2019.
Éric Fusy, Benjamin Lévêque, Orientations and bijections for toroidal maps with prescribed face-degrees and essential girth, arXiv:1807.00522 [math.CO], 2018. See Proposition 25 p. 37.

Cf. A002293, A308524, A308526, A289208, A006422.
Cf. A000302, A036829.
Cf. A386565, A386611, A386612.

n:=20:
dev_A := series(RootOf(A-x*(1+A)^4, A), x = 0, n+1):
seq(coeff(series(subs(A=dev_A, A/(1-3*A)^2), x, n+1), x, k), k=0..n);

terms = 21;
A[] = 0; Do[A[x] = x (1 + A[x])^4 + O[x]^terms, terms];
CoefficientList[A[x]/(1 - 3 A[x])^2, x] (* Jean-François Alcover, Jun 17 2019 *)

my(N=30, x='x+O('x^N), g=x*sum(k=0, N, binomial(4*k+2, k)/(k+1)*x^k)); concat(0, Vec(g*(1-g)/(1-4*g)^2)) \\ Seiichi Manyama, Jul 19 2025

G.f.: A/(1-3*A)^2 where A=x(1+A)^4 is the g.f. of A002293.
From Vaclav Kotesovec, Jun 25 2019: (Start)
Recurrence: 81*(n-1)*(3*n - 2)*(3*n - 1)*(24*n - 37)*a(n) = 24*(13824*n^4 - 59328*n^3 + 92832*n^2 - 62278*n + 14653)*a(n-1) - 2048*(2*n - 3)*(4*n - 7)*(4*n - 5)*(24*n - 13)*a(n-2).
a(n) ~ 2^(8*n - 3) / 3^(3*n). (End)
From Seiichi Manyama, Jul 19 2025: (Start)
G.f.: g*(1-g)/(1-4*g)^2 where g*(1-g)^3 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/4) * log( Sum_{k>=0} binomial(4*k,k)*x^k ). (End)
From Seiichi Manyama, Jul 28 2025: (Start)
a(n) = Sum_{k=0..n-1} binomial(4*k-2+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-1,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k-1,k). (End)

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).

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).

A386611 a(n) = Sum_{k=0..n-1} binomial(4k,k) binomial(4n-4k,n-k-1).

Original entry on oeis.org

0, 1, 12, 126, 1268, 12513, 122148, 1184364, 11432100, 109997460, 1055891248, 10117633542, 96812495820, 925334377822, 8836315646616, 84317468847768, 804064275489924, 7663595943744876, 73009005101019792, 695263276434909976, 6618709687608909648, 62989317586872238689

Offset: 0

Seiichi Manyama, Jul 27 2025

nonn

Cf. A308523, A386565, A386612.

Cf. A008549, A386613, A386615.

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

G.f.: g^2 * (g-1)/(4-3*g)^2 where g=1+x*g^4.
G.f.: g/((1-g) * (1-4*g)^2) where g*(1-g)^3 = x.
a(n) = Sum_{k=0..n-1} binomial(4*k+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+1,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k+1,k).

A385251 a(n) = Sum_{k=0..n-1} binomial(4k-3,k) binomial(4n-4k,n-k-1).

Original entry on oeis.org

0, 1, 9, 84, 790, 7452, 70401, 665692, 6298236, 59612556, 564393460, 5344664400, 50621130078, 479513718116, 4542730477758, 43039907282664, 407809863233592, 3864303038901996, 36619104142640460, 347027703183853552, 3288802989845088504, 31169274939274755312

Offset: 0

Seiichi Manyama, Jul 28 2025

nonn

Cf. A078995, A308523, A386565, A386611, A386612.

a(n) = sum(k=0, n-1, binomial(4*k-3, k)*binomial(4*n-4*k, n-k-1));

G.f.: (g-1)/(g * (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-3+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).

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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A308523 Number of essentially simple rooted toroidal triangulations with n vertices.

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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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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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A386611 a(n) = Sum_{k=0..n-1} binomial(4*k,k) * binomial(4*n-4*k,n-k-1).

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A385251 a(n) = Sum_{k=0..n-1} binomial(4*k-3,k) * binomial(4*n-4*k,n-k-1).

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

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).

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

A386611 a(n) = Sum_{k=0..n-1} binomial(4k,k) binomial(4n-4k,n-k-1).

A385251 a(n) = Sum_{k=0..n-1} binomial(4k-3,k) binomial(4n-4k,n-k-1).