A036829 -id:A036829 - cp's OEIS frontend

A036917 G.f.: (4/Pi^2)EllipticK(4x^(1/2))^2.

1, 8, 88, 1088, 14296, 195008, 2728384, 38879744, 561787864, 8206324928, 120929313088, 1794924383744, 26802975999424, 402298219288064, 6064992788397568, 91786654611673088, 1393772628452578264, 21227503080738294464, 324160111169327247424

Offset: 0

N. J. A. Sloane

			G.f. = 1 + 8*x + 88*x^2 +  1088*x^3 + 14296*x^5 + 195008*x^5 + ... - _Michael Somos_, May 29 2023

M. Petkovsek et al., "A=B", Peters, p. ix of second printing.

Reinhard Zumkeller, Table of n, a(n) for n = 0..500
B. Adamczewski, J. P. Bell, and E. Delaygue, Algebraic independence of G-functions and congruences "a la Lucas", arXiv preprint arXiv:1603.04187 [math.NT], 2016.
E. Delaygue, Arithmetic properties of Apery-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013.
Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
Ji-Cai Liu and He-Xia Ni, Supercongruences for Almkvist--Zudilin sequences, arXiv:2004.07652 [math.NT], 2020. See Vn.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020. See Vn.
Zhi-Hong Sun, Congruences for two types of Apery-like sequences, arXiv:2005.02081 [math.NT], 2020.

Cf. A002894, A036915, A057703.
Cf. A007318, A036916, A036829.
The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

a036917 n = sum $ map
   (\k -> (a007318 (2*n-2*k) (n-k))^2 * (a007318 (2*k) k)^2) [0..n]
-- Reinhard Zumkeller, May 24 2012

a[n_] := (16 (n - 1/2)(2*n^2 - 2*n + 1)a[n - 1] - 256(n - 1)^3 a[n - 2])/n^3; a[0] = 1; a[1] = 8; Array[a, 19, 0] (* Or *)
f[n_] := Sum[(Binomial[2 (n - k), n - k] Binomial[2 k, k])^2, {k, 0, n}]; Array[f, 19, 0] (* Or *)
lmt = 20; Take[ 4^Range[0, 2 lmt]*CoefficientList[ Series[(4/Pi^2) EllipticK[4 x^(1/2)]^2, {x, 0, lmt}], x^(1/2)], lmt] (* Robert G. Wilson v *)
a[n_] := HypergeometricPFQ[{1/2, 1/2, -n, -n}, {1, 1/2-n, 1/2-n}, 1] * 4^n * (2n-1)!!^2 / n!^2 (* Vladimir Reshetnikov, Mar 08 2014 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[3, 0, EllipticNomeQ[16*x]]^4, {x, 0, n}]; (* Michael Somos, May 30 2023 *)

for(n=0,25, print1(sum(k=0,n, (binomial(2*n-2*k,n-k) *binomial(2*k,k))^2), ", ")) \\ G. C. Greubel, Oct 24 2017

a(n) = if(n<0, 0, polcoeff(agm(1, sqrt(1 - 16*x + x*O(x^n)))^-2, n)); /* Michael Somos, May 29 2023 */

a(n) = (16*(n-1/2)*(2*n^2-2*n+1)*a(n-1)-256*(n-1)^3*a(n-2))/n^3.
a(n) = Sum_{k=0..n} (C(2 * (n-k), n-k) * C(2 * k, k))^2. [corrected by Tito Piezas III, Oct 19 2010]
a(n) = hypergeom([1/2, 1/2, -n, -n], [1, 1/2-n, 1/2-n], 1) * 4^n * (2n-1)!!^2 / n!^2. - Vladimir Reshetnikov, Mar 08 2014
a(n) ~ 2^(4*n+1) * log(n) / (n*Pi^2) * (1 + (4*log(2) + gamma)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Nov 28 2015
G.f. y=A(x) satisfies: 0 = x^2*(16*x - 1)^2*y''' + 3*x*(16*x - 1)*(32*x - 1)*y'' + (1792*x^2 - 112*x + 1)*y' + 8*(32*x - 1)*y. - Gheorghe Coserea, Jul 03 2018
G.f.: 1 / AGM(1, sqrt(1 - 16*x))^2. - Vaclav Kotesovec, Oct 01 2019
It appears that a(n) is equal to the coefficient of (x*y*z*t)^n in the expansion of (1+x+y+z-t)^n * (1+x+y-z+t)^n * (1+x-y+z+t)^n * (1-x+y+z+t)^n. Cf. A000172. - Peter Bala, Sep 21 2021
G.f. y = A(x) satisfies 0 = x*(1 - 16*x)*(2*y''*y - y'*y') + 2*(1 - 32*x)*y*y' - 16*y*y. - Michael Somos, May 29 2023
Expansion of theta_3(0, q)^4 in powers of m/16 where the modulus m = k^2. - Michael Somos, May 30 2023
From Paul D. Hanna, Mar 25 2024: (Start)
G.f. ( Sum_{n>=0} binomial(2*n,n)^2 * x^n )^2.
G.f. Sum_{n>=0} binomial(2*n,n)^3 * x^n * (1 - 16*x)^n. (End)

Replaced complicated definition via a formula with simple generating function provided by Vladeta Jovovic, Dec 01 2003. Thanks to Paul D. Hanna for suggesting this. - N. J. A. Sloane, Mar 25 2024

A006256 a(n) = Sum_{k=0..n} binomial(3k,k)binomial(3n-3k,n-k).

Original entry on oeis.org

1, 6, 39, 258, 1719, 11496, 77052, 517194, 3475071, 23366598, 157206519, 1058119992, 7124428836, 47983020624, 323240752272, 2177956129818, 14677216121871, 98923498131762, 666819212874501, 4495342330033938, 30308036621747679, 204356509814519712

Offset: 0

N. J. A. Sloane, Don Knuth

The right-hand sides of several of the "Ruehr identities". - N. J. A. Sloane, Feb 20 2020

Convolution of A005809 with itself. - Emeric Deutsch, May 22 2003

Allouche, J-P. "Two binomial identities of Ruehr Revisited." The American Mathematical Monthly 126.3 (2019): 217-225.
Alzer, Horst, and Helmut Prodinger. "On Ruehr's Identities." Ars Comb. 139 (2018): 247-254.
Bai, Mei, and Wenchang Chu. "Seven equivalent binomial sums." Discrete Mathematics 343.2 (2020): 111691.
M. Petkovsek et al., A=B, Peters, 1996, p. 165.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Jean-Paul Allouche, A generalization of an identity due to Kimura and Ruehr, arXiv preprint arXiv:1706.08929 [math.NT], 2017.
Jean-Paul Allouche, Two exercises of Comtet and two identities of Ruehr, arXiv preprint arXiv:1707.05751 [math.NT], 2017.
Rui Duarte and António Guedes de Oliveira, Short note on the convolution of binomial coefficients, arXiv:1302.2100 [math.CO], 2013 and J. Int. Seq. 16 (2013) #13.7.6.
Shalosh B. Ekhad, Doron Zeilberger, Some Remarks on a recent article by J.-P. Allouche, arXiv:1903.09511 [math.CO], 2019.
N, Kimura and O. G. Ruehr, Change of variable formula for definite integral. Problem E2765, Am. Math. Mnthly, 87, 1980, 307-308.
S. Meehan, A. Tefera, M. Weselcouch, A. Zeleke, Proofs of Ruehr's identities, Integers 14 (2014) A10.
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Y_n for s=3).

Cf. A036829.
Cf. A005809, A160906, A183160, A226751.
Cf. A000302, A078995, A079563, A079678, A079679.
Cf. A141223, A385605, A385632.

a006256 n = a006256_list !! n
a006256_list = f (tail a005809_list) [1] where
   f (x:xs) zs = (sum $ zipWith (*) zs a005809_list) : f xs (x : zs)
-- Reinhard Zumkeller, Sep 21 2014

[&+[Binomial(3*k, k) *Binomial(3*n-3*k, n-k): k in [0..n]]:n in  [0..22]]; // Vincenzo Librandi, Feb 21 2020

a:= proc(n) option remember; `if`(n<2, 5*n+1,
      ((216*n^2-270*n+96) *a(n-1)
      -81*(3*n-2)*(3*n-4) *a(n-2)) /(n*(16*n-8)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 07 2012

a[n_] := HypergeometricPFQ[{1/3, 2/3, 1/2-n, -n}, {1/2, 1/3-n, 2/3-n}, 1]*(3n)!/(n!*(2n)!); Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 20 2012 *)
Table[Sum[Binomial[3k,k]Binomial[3n-3k,n-k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Oct 23 2013 *)

a(n)=sum(k=0,n, binomial(3*k,k)*binomial(3*n-3*k,n-k)) \\ Charles R Greathouse IV, Feb 07 2017

a = lambda n: binomial(3*n+1,n)*hypergeometric([1,-n],[2*n+2],-2)
[simplify(a(n)) for n in range(20)] # Peter Luschny, May 19 2015

a(n) = (3/4)*(27/4)^n*(1+c/sqrt(n)+o(n^(-1/2))) where c = (2/3)*sqrt(1/(3*Pi)) = 0.217156671956853298... More generally, a(n, m) = sum(k=0, n, C(m*k,k) *C(m*(n-k),n-k)) is asymptotic to (1/2)*m/(m-1)*(m^m/(m-1)^(m-1))^n. See A000302, A078995 for cases m=2 and 4. - Benoit Cloitre, Jan 26 2003, extended by Vaclav Kotesovec, Nov 06 2012
G.f.: 1/(1-3*z*g^2)^2, where g=g(z) is given by g=1+z*g^3, g(0)=1, i.e. (in Maple command) g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z). - Emeric Deutsch, May 22 2003
D-finite with recurrence: 6*(36*n^2-45*n+16)*a(n-1) - 81*(3*n-4)*(3*n-2)*a(n-2) - 8*n*(2*n-1)*a(n) = 0. - Vaclav Kotesovec, Oct 05 2012
From Rui Duarte and António Guedes de Oliveira, Feb 17 2013: (Start)
a(n) = sum(k=0, n, C(3*k+l,k)*C(3*(n-k)-l,n-k)) for every real number l.
a(n) = sum(k=0, n, 2^(n-k)*C(3n+1,k)).
a(n) = sum(k=0, n, 3^(n-k)*C(2n+k,k)).  (End)
From Akalu Tefera, Sean Meehan, Michael Weselcouch, and Aklilu Zeleke, May 11 2013: (Start)
a(n) = sum(k=0, 2n, (-3)^k*C(3n - k, n)).
a(n) = sum(k=0, 2n, (-4)^k*C(3n + 1, 2n - k)).
a(n) = sum(k=0, n, 3^k*C(3n - k, 2n)).
a(n) = sum(k=0, n, 2^k*C(3n + 1, n - k)).  (End)
a(n) = C(3*n+1,n)*Hyper2F1(1,-n,2*n+2,-2). - Peter Luschny, May 19 2015
a(n) = [x^n] 1/((1-3*x) * (1-x)^(2*n+1)). - Seiichi Manyama, Aug 03 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k). - Seiichi Manyama, Aug 07 2025

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)

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)

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).

A386368 a(n) = Sum_{k=0..n-1} binomial(6k,k) binomial(6n-6k-2,n-k-1).

Original entry on oeis.org

0, 1, 16, 246, 3736, 56421, 849432, 12763878, 191548464, 2871970110, 43031833656, 644432826478, 9646983339456, 144366433138955, 2159869510669320, 32306874783230556, 483151884326658144, 7224464127509984490, 108011596038055519680, 1614676987907480393940

Offset: 0

Seiichi Manyama, Jul 19 2025

nonn

			(1/6) * log( Sum_{k>=0} binomial(6*k,k)*x^k ) = x + 8*x^2 + 82*x^3 + 934*x^4 + 56421*x^5/5 + ...

Cf. A000302, A036829, A308523, A386367.
Cf. A079679, A386567, A386615, A386616.
Cf. A002295, A130564.

A386368 := proc(n::integer)
    add(binomial(6*k,k)*binomial(6*n-6*k-2,n-k-1),k=0..n-1) ;
end proc:
seq(A386368(n),n=0..80) ; # R. J. Mathar, Jul 30 2025

a(n) = sum(k=0, n-1, binomial(6*k, k)*binomial(6*n-6*k-2, n-k-1));

my(N=20, x='x+O('x^N), g=x*sum(k=0, N, binomial(6*k+4, k)/(k+1)*x^k)); concat(0, Vec(g*(1-g)/(1-6*g)^2))

G.f.: g*(1-g)/(1-6*g)^2 where g*(1-g)^5 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/6) * log( Sum_{k>=0} binomial(6*k,k)*x^k ).
G.f.: (g-1)/(6-5*g)^2 where g=1+x*g^6.
a(n) = Sum_{k=0..n-1} binomial(6*k-2+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-1,k).
a(n) = Sum_{k=0..n-1} 6^(n-k-1) * binomial(5*n+k-1,k).
Conjecture D-finite with recurrence 48828125*(n-1)*(5*n-4)*(5*n-3) *(432862082629612805*n -769306661967834399) *(5*n-2)*(5*n-1)*a(n) +1125000*(-405245406115816219575000*n^6 +2613180799468910510392500*n^5 -7667164406968651479521250*n^4 +13834502135358262506660375*n^3 -16251583347734702117341345*n^2 +11251247074043948959380314*n -3395699069351241765495720)*a(n-1) +33592320*(142281690918326440537500*n^6 -1266424338521609272012500*n^5 +5236041263583271687953750*n^4 -12786608152035075786775875*n^3 +18838556229131595646260055*n^2 -15323925851720394901667853*n +5240681406952416812161236)*a(n-2) -53444359913472*(6*n-17) *(395547729523405*n -538181211711288)*(6*n-13) *(3*n-7)*(2*n-5) *(3*n-8)*a(n-3)=0. - R. J. Mathar, Jul 30 2025

A386367 a(n) = Sum_{k=0..n-1} binomial(5k,k) binomial(5n-5k-2,n-k-1).

Original entry on oeis.org

0, 1, 13, 163, 2021, 24930, 306655, 3765448, 46182101, 565939603, 6931070490, 84845250370, 1038235255415, 12700966517968, 155336699256808, 1899439862390640, 23222289820948405, 283872591297526505, 3469680960837171415, 42404345427419774621, 518193229118757697930

Offset: 0

Seiichi Manyama, Jul 19 2025

nonn

			(1/5) * log( Sum_{k>=0} binomial(5*k,k)*x^k ) = x + 13*x^2/2 + 163*x^3/3 + 2021*x^4/4 + 4986*x^5 + ...

Cf. A000302, A036829, A308523, A386368.
Cf. A079678, A386566, A386613, A386614.
Cf. A002294, A118971.

a(n) = sum(k=0, n-1, binomial(5*k, k)*binomial(5*n-5*k-2, n-k-1));

my(N=30, x='x+O('x^N), g=x*sum(k=0, N, binomial(5*k+3, k)/(k+1)*x^k)); concat(0, Vec(g*(1-g)/(1-5*g)^2))

G.f.: g*(1-g)/(1-5*g)^2 where g*(1-g)^4 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/5) * log( Sum_{k>=0} binomial(5*k,k)*x^k ).
G.f.: (g-1)/(5-4*g)^2 where g=1+x*g^5.
a(n) = Sum_{k=0..n-1} binomial(5*k-2+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-1,k).
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(4*n+k-1,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)

cp's OEIS Frontend

A036917 G.f.: (4/Pi^2)*EllipticK(4*x^(1/2))^2.

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A006256 a(n) = Sum_{k=0..n} binomial(3*k,k)*binomial(3*n-3*k,n-k).

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

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

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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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A386368 a(n) = Sum_{k=0..n-1} binomial(6*k,k) * binomial(6*n-6*k-2,n-k-1).

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A386367 a(n) = Sum_{k=0..n-1} binomial(5*k,k) * binomial(5*n-5*k-2,n-k-1).

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A036917 G.f.: (4/Pi^2)EllipticK(4x^(1/2))^2.

A006256 a(n) = Sum_{k=0..n} binomial(3k,k)binomial(3n-3k,n-k).

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

A386368 a(n) = Sum_{k=0..n-1} binomial(6k,k) binomial(6n-6k-2,n-k-1).

A386367 a(n) = Sum_{k=0..n-1} binomial(5k,k) binomial(5n-5k-2,n-k-1).