A006256 -id:A006256 - cp's OEIS frontend

A385632 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(5*n+1,k).

1, 8, 81, 872, 9669, 109128, 1246419, 14359304, 166512285, 1940885504, 22717923586, 266833238328, 3143237113479, 37119019790016, 439290932937672, 5208668386199112, 61861932606093901, 735804601177846968, 8763478151940329859, 104498114621004830160, 1247410783999193335434

Offset: 0

Seiichi Manyama, Aug 03 2025

nonn

Cf. A006256, A141223, A385605.
Cf. A001449, A079589, A079678, A371753, A386812.
Cf. A386699.

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

a(n) = [x^n] 1/((1-3*x) * (1-x)^(4*n+1)).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(4*n+k,k).
a(n) = 3^(5*n+1)*2^(-4*n-1) - binomial(5*n+1, n)*(hypergeom([1, -1-4*n], [1+n], -1/2) - 1). - Stefano Spezia, Aug 05 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(5*n+1,k) * binomial(5*n-k,n-k). - Seiichi Manyama, Aug 07 2025
G.f.: g^2/((3-2*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 14 2025
From Seiichi Manyama, Aug 16 2025: (Start)
G.f.: 1/(1 - x*g^3*(15-7*g)) where g = 1+x*g^5 is the g.f. of A002294.
G.f.: B(x)^2/(1 + 2*(B(x)-1)/5), where B(x) is the g.f. of A001449. (End)

A226751 G.f.: 1 / (1 + 6xG(x) - 7xG(x)^2), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 9, 48, 289, 1761, 10932, 68664, 435201, 2777763, 17829489, 114968052, 744178716, 4832624044, 31469746632, 205422018288, 1343734578561, 8806130111847, 57805893969531, 380013533789928, 2501507255441049, 16486378106441697, 108773240389894056

Offset: 0

Paul D. Hanna, Jun 16 2013

nonn

			G.f.: A(x) = 1 + x + 9*x^2 + 48*x^3 + 289*x^4 + 1761*x^5 + 10932*x^6 +...
A related series is G(x) = 1 + x*G(x)^3, where
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
G(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
such that A(x) = 1/(1 + 6*x*G(x) - 7*x*G(x)^2).

Vincenzo Librandi and Joerg Arndt, Table of n, a(n) for n = 0..200

Cf. A147855, A001764.
Cf. A005809, A006256, A160906, A183160.
Cf. A226705, A226733.

Table[Sum[Binomial[n+2*k,n-k]*Binomial[2*n-2*k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 17 2013 *)

{a(n)=sum(k=0, n, binomial(n+2*k, n-k)*binomial(2*n-2*k, k))}
for(n=0, 30, print1(a(n), ", "))

{a(n)=sum(k=0, n, binomial(2*k, n-k)*binomial(3*n-2*k, k))}
for(n=0, 30, print1(a(n), ", "))

{a(n)=local(G=1+x); for(i=0, n, G=1+x*G^3+x*O(x^n)); polcoeff(1/(1+6*x*G-7*x*G^2), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=local(G=1+x); for(i=0, n,G=1+x*G^3+x*O(x^n)); polcoeff(1/(1-x*G-7*x^2*G^4), n)}
for(n=0, 30, print1(a(n), ", "))

a(n) = Sum_{k=0..n} C(2*k, n-k) * C(3*n-2*k, k).
a(n) = Sum_{k=0..n} C(n+2*k, n-k) * C(2*n-2*k, k).
a(n) = Sum_{k=0..n} C(2*n+2*k, n-k) * C(n-2*k, k).
a(n) = Sum_{k=0..n} C(3*n+2*k, n-k) * C(-2*k, k).
G.f.: 1/(1 - x*G(x) - 7*x^2*G(x)^4), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
a(n) ~ 3^(3*n+3/2)/(5*sqrt(Pi*n)*2^(2*n+1)). - Vaclav Kotesovec, Jun 17 2013
Conjecture: 18*n*(2*n-1)*(55*n-76)*a(n) +(-11605*n^3+28521*n^2-20870*n+4536)*a(n-1) -24*(55*n-21)*(3*n-4)*(3*n-2)*a(n-2)=0. - R. J. Mathar, Jun 14 2016
From Seiichi Manyama, Aug 05 2025: (Start)
a(n) = [x^n] 1/((1+2*x) * (1-x)^(2*n+1)).
a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(3*n+1,k).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(2*n+k,k). (End)
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k).
G.f.: G(x)^2/((-2+3*G(x)) * (3-2*G(x))) where G(x) = 1+x*G(x)^3 is the g.f. of A001764. (End)
G.f.: B(x)^2/(1 + 5*(B(x)-1)/3), where B(x) is the g.f. of A005809. - Seiichi Manyama, Aug 15 2025

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).

A079563 a(n) = a(n,m) = Sum_{k=0..n} binomial(mk,k)binomial(m*(n-k),n-k) for m=7.

Original entry on oeis.org

1, 14, 231, 3934, 67851, 1177974, 20531770, 358788696, 6281076123, 110103674128, 1931983053056, 33926800240578, 596145343139514, 10480467311987778, 184327560283768776, 3243034966775972144, 57074433199551436347

Offset: 0

Benoit Cloitre, Jan 26 2003

nonn

More generally, for m>=2, a(n,m) = Sum_{k=0..n} binomial(m*k,k)*binomial(m*(n-k),n-k) is asymptotic to 1/2*m/(m-1)*(m^m/(m-1)^(m-1))^n * (1 + (2*m-4)/(3*sqrt(Pi*n*m*(m-1)/2))), extended by Vaclav Kotesovec, May 25 2020

See A000302, A006256, A078995 for cases m=2,3 and 4.

Seiichi Manyama, Table of n, a(n) for n = 0..802
Rui Duarte and António Guedes de Oliveira, Short note on the convolution of binomial coefficients, arXiv:1302.2100 [math.CO], 2013.
D. Merlini, R. Sprugnoli, and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344.

Cf. A006256, A078995, A079678, A079679.

Cf. A002296.

a(n) = (7/12)*(823543/46656)^n*(1+c/sqrt(n)+o(n^-1/2)) where c=0.41...
c = 10/(3*sqrt(21*Pi)) = 0.410387535383... - Vaclav Kotesovec, May 25 2020
From Rui Duarte and António Guedes de Oliveira, Feb 17 2013: (Start)
a(n) = Sum_{k=0..n} binomial(7*k+x,k)*binomial(7*(n-k)-x,n-k) for any real x.
a(n) = Sum_{k=0..n} 6^(n-k)*binomial(7*n+1,k).
a(n) = Sum_{k=0..n} 7^(n-k)*binomial(6*n+k,k). (End)
a(n) = [x^n] 1/((1-7*x) * (1-x)^(6*n+1)). - Seiichi Manyama, Aug 03 2025
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} 7^k * (-6)^(n-k) * binomial(7*n+1,k) * binomial(7*n-k,n-k).
G.f.: g^2/(7-6*g)^2 where g = 1+x*g^7 is the g.f. of A002296. (End)

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)

A188911 Binomial convolution of the binomial coefficients bin(3n,n) (A005809).

Original entry on oeis.org

1, 6, 48, 438, 4356, 46056, 509106, 5814738, 68050116, 811240872, 9810384048, 119990105208, 1481115683754, 18421300391760, 230574816629310, 2901721280735838, 36688485233689668, 465774244616805624, 5934465567864915024

Offset: 0

Emanuele Munarini, Apr 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..90

Cf. A005809, A001764, A006256, A006013, A045721, A188912, A188913.

Table[Sum[Binomial[n,k]Binomial[3k,k]Binomial[3n-3k,n-k], {k,0,n}], {n,0,22}]

makelist(sum(binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k),k,0,n),n,0,12);

a(n)=sum(k=0,n,binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k));
vector(66,n,a(n-1)) /* show terms */ /* Joerg Arndt, Apr 13 2011 */

a(n) = Sum_{k=0..n} binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k).
E.g.f.: F(1/3,2/3;1/2,1;27*x/4)^2, where F(a1,a2;b1,b2;z) is a hypergeometric series.
Recurrence: 8*n^2 * (2*n-1)^2 * (9*n^3 - 54*n^2 + 102*n - 61)*a(n) = 24*(3*n-1)*(108*n^6 - 855*n^5 + 2628*n^4 - 4059*n^3 + 3380*n^2 - 1470*n + 264)*a(n-1) - 18*(3645*n^7 - 34992*n^6 + 138348*n^5 - 291843*n^4 + 352980*n^3 - 241794*n^2 + 84684*n - 11104)*a(n-2) + 2187*(n-2)^2 * (3*n-7)*(3*n-5)*(9*n^3 - 27*n^2 + 21*n - 4)*a(n-3). - Vaclav Kotesovec, Feb 25 2014
a(n) ~ 3^(3*n+1) / (Pi * n * 2^(n+1)). - Vaclav Kotesovec, Feb 25 2014

A188912 Binomial convolution of the binomial coefficients bin(3n,n)/(2n+1) (A001764).

Original entry on oeis.org

1, 2, 8, 42, 260, 1816, 13962, 116094, 1029124, 9609144, 93569808, 942642696, 9763181946, 103455616400, 1117379189926, 12264816349938, 136501928050116, 1537591374945704, 17503603786398576, 201128739609458904, 2330480521265639136

Offset: 0

Emanuele Munarini, Apr 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..93

Cf. A005809, A001764, A005809, A006256, A006013, A045721, A188911, A188913

Table[Sum[Binomial[n,k]Binomial[3k,k]/(2k+1)Binomial[3n-3k,n-k]/(2n-2k+1), {k,0,n}], {n,0,22}]

makelist(sum(binomial(n,k)*binomial(3*k,k)/(2*k+1)*binomial(3*n-3*k,n-k)/(2*n-2*k+1),k,0,n),n,0,12);

a(n) = Sum_{k=0..n} binomial(n, k)*binomial(3*k, k)*binomial(3*n-3*k, n-k)/((2*k+1)*(2*n-2*k+1)).
E.g.f.: F(1/3,2/3;1,3/2;27*x/4)^2, where F(a1,a2;b1,b2;z) is a hypergeometric series.
From Vaclav Kotesovec, Jun 10 2019: (Start)
Recurrence: 8*n^2*(n+1)*(2*n+1)^2*(9*n^3-54*n^2+84*n-35)*a(n) = 24*n*(324*n^7-2187*n^6+4689*n^5-4185*n^4+1464*n^3+122*n^2-223*n+44)*a(n-1) - 18*(n-1)*(3645*n^7-30618*n^6+96066*n^5-144585*n^4+103662*n^3-21834*n^2-10860*n+4480)*a(n-2) + 2187*(n-2)^2*(n-1)*(3*n-7)*(3*n-5)*(9*n^3-27*n^2+3*n+4)*a(n-3).
a(n) ~ 3^(3*n + 1) / (Pi * n^3 * 2^(n + 1)). (End)

A188913 Binomial convolution of the binomial coefficients bin(3n,n) (A005809) and bin(3n,n)/(2n+1) (A001764).

Original entry on oeis.org

1, 4, 24, 168, 1300, 10896, 97734, 928752, 9262116, 96091440, 1029267888, 11311712352, 126921365298, 1448378629600, 16760687848890, 196237061599008, 2320532776851972, 27676644749022672, 332568471941572944, 4022574792189178080

Offset: 0

Emanuele Munarini, Apr 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..105

Cf. A005809, A001764, A006256, A006013, A045721, A188911, A188912.

Table[Sum[Binomial[n,k]Binomial[3k,k]Binomial[3n-3k,n-k]/(2n-2k+1), {k,0,n}], {n,0,22}]

makelist(sum(binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k)/(2*n-2*k+1),k,0,n),n,0,12);

a(n) = sum(k=0,n,binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k)/(2*n-2*k+1));
vector(66, n, a(n-1)) /* show terms */ /* Joerg Arndt, Apr 13 2011 */

a(n) = sum(binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k)/(2*n-2*k+1),k=0..n).
E.g.f.: F(1/3,2/3;1/2,1;27*x/4)*F(1/3,2/3;1,3/2;27*x/4), where F(a1,a2;b1,b2;z) is a hypergeometric series.
Recurrence: 8*n^2 * (2*n+1)^2 * (9*n^3 - 54*n^2 + 84*n - 35)*a(n) = 24*(324*n^7 - 2187*n^6 + 4689*n^5 - 4185*n^4 + 1464*n^3 + 122*n^2 - 223*n + 44)*a(n-1) - 18*(3645*n^7 - 30618*n^6 + 96066*n^5 - 144585*n^4 + 103662*n^3 - 21834*n^2 - 10860*n + 4480)*a(n-2) + 2187*(n-2)*(n-1)*(3*n-7)*(3*n-5)*(9*n^3 - 27*n^2 + 3*n + 4)*a(n-3). - Vaclav Kotesovec, Feb 25 2014
a(n) ~ 3^(3*n+1) / (Pi*n^2*2^(n+1)). - Vaclav Kotesovec, Feb 25 2014

A358050 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(kj,j) binomial(k*(n-j),n-j).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 6, 16, 4, 0, 1, 8, 39, 64, 5, 0, 1, 10, 72, 258, 256, 6, 0, 1, 12, 115, 664, 1719, 1024, 7, 0, 1, 14, 168, 1360, 6184, 11496, 4096, 8, 0, 1, 16, 231, 2424, 16265, 57888, 77052, 16384, 9, 0, 1, 18, 304, 3934, 35400, 195660, 543544, 517194, 65536, 10, 0

Offset: 0

Seiichi Manyama, Oct 31 2022

			Square array begins:
  1, 1,    1,     1,     1,      1, ...
  0, 2,    4,     6,     8,     10, ...
  0, 3,   16,    39,    72,    115, ...
  0, 4,   64,   258,   664,   1360, ...
  0, 5,  256,  1719,  6184,  16265, ...
  0, 6, 1024, 11496, 57888, 195660, ...

Column k=0-7 give: A000007, A001477(n+1), A000302, A006256, A078995, A079678, A079679, A079563.
Main diagonal gives A358145.
Cf. A358146.

T(n, k) = sum(j=0, n, binomial(k*j, j)*binomial(k*(n-j), n-j));

T(n, k) = sum(j=0, n, (k-1)^(n-j)*binomial(k*n+1, j));

T(n, k) = sum(j=0, n, k^(n-j)*binomial((k-1)*n+j, j));

T(n,k) = Sum_{j=0..n} (k-1)^(n-j) * binomial(k*n+1,j).

T(n,k) = Sum_{j=0..n} k^(n-j) * binomial((k-1)*n+j,j).

A378504 Expansion of (Sum_{k>=0} binomial(3k,k) x^k)^4.

Original entry on oeis.org

1, 12, 114, 984, 8055, 63744, 492702, 3742704, 28053423, 208057260, 1529802648, 11168142048, 81041199876, 585045970992, 4204705925670, 30101448952032, 214756404746031, 1527491122906212, 10834911076417458, 76666402505673720, 541277205506059743

Offset: 0

Seiichi Manyama, Nov 28 2024

nonn

Marko Riedel et al, Identity between two generating functions

Cf. A005809, A006256, A378483.

Cf. A002697, A378484.

nmax = 20; CoefficientList[Series[Sum[Binomial[3*k, k] * x^k, {k, 0, nmax}]^4, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 23 2025 *)

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, binomial(3*k, k)*x^k)^4)

a(n) = Sum_{i+j+k+l=n, i,j,k,l >= 0} binomial(3*i,i) * binomial(3*j,j) * binomial(3*k,k) * binomial(3*l,l).
G.f.: B(x)^4 where B(x) is the g.f. of A005809.
4*a(n) - 27*a(n-1) = 3*A006256(n) + A005809(n) for n > 0.
Sum_{n >= 0} a(n) * z^n / (1+z)^(3*n) = (1+z)^4 / (1-2*z)^4. - Marko Riedel, Jul 22 2025
From Vaclav Kotesovec, Jul 23 2025: (Start)
Recurrence: 8*(n-1)*n*(2*n - 1)*a(n) = 6*(n-1)*(36*n^2 - 9*n - 5)*a(n-1) - 81*n*(3*n - 2)*(3*n - 1)*a(n-2).
a(n) ~ n * 3^(3*n+2) / 2^(2*n+4). (End)

cp's OEIS Frontend

A385632 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(5*n+1,k).

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A226751 G.f.: 1 / (1 + 6*x*G(x) - 7*x*G(x)^2), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

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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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A079563 a(n) = a(n,m) = Sum_{k=0..n} binomial(m*k,k)*binomial(m*(n-k),n-k) for m=7.

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

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Maple

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A188911 Binomial convolution of the binomial coefficients bin(3n,n) (A005809).

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Maxima

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A188912 Binomial convolution of the binomial coefficients bin(3n,n)/(2n+1) (A001764).

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Maxima

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A188913 Binomial convolution of the binomial coefficients bin(3n,n) (A005809) and bin(3n,n)/(2n+1) (A001764).

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Mathematica

Maxima

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A226751 G.f.: 1 / (1 + 6xG(x) - 7xG(x)^2), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

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

A079563 a(n) = a(n,m) = Sum_{k=0..n} binomial(mk,k)binomial(m*(n-k),n-k) for m=7.

A358050 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(kj,j) binomial(k*(n-j),n-j).

A378504 Expansion of (Sum_{k>=0} binomial(3k,k) x^k)^4.