A066380 -id:A066380 - cp's OEIS frontend

A066381 a(n) = Sum_{k=0..n} binomial(4*n,k).

1, 5, 37, 299, 2517, 21700, 190051, 1683218, 15033173, 135142796, 1221246132, 11083374659, 100946732307, 922205369324, 8446802334994, 77542088287444, 713250450657109, 6572130378649468, 60652194138406780, 560522209086365852

Offset: 0

N. J. A. Sloane, Dec 23 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 0..150

Cf. A002293, A032443, A066380, A371739.

ogf := eval(s/((s-2)*(3*s-4)), s = RootOf(1-s+x*s^4, s));
series(ogf, x=0, 25); # Mark van Hoeij, May 05 2013

Table[Sum[Binomial[4*n,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 03 2015 *)

a[0]:1$ a[1]:5$ a[n]:=8*((3784*n^6-18764*n^5+34432*n^4 -28138*n^3+9028*n^2-24*n-315)*a[n-1]+16*(3-2*n)*(4*n-5)*(4*n-7)*(44*n^3-34*n^2-2*n+3)*a[n-2])/(3*n*(3*n-1)*(3*n-2)*(44*n^3-166*n^2 +198*n-73))$ makelist(a[n],n,0,1000); /* Tani Akinari, Sep 02 2014 */

{ for (n=0, 150, a=0; for (k=0, n, a+=binomial(4*n, k)); write("b066381.txt", n, " ", a) ) } \\ Harry J. Smith, Feb 12 2010

G.f.: s/((s-2)*(3*s-4)) where s = o.g.f.(A002293) which satisfies 1-s+x*s^4 = 0. - Mark van Hoeij, May 05 2013
a(0) = 1, a(n) = 16*a(n-1)-2*(44*n^3-34*n^2-2*n+3)*(4*n-4)!/(n!*(3*n-1)!). - Tani Akinari, Sep 02 2014
a(n) are special values of the hypergeometric function 2F1, in Maple notation: a(n)=16^n-binomial(4*n,n+1)*hypergeom([1,-3*n+1],[n+2],-1), n=0,1,... . - Karol A. Penson, Jun 03 2015
a(n) ~ (256/27)^n * sqrt(3/(2*Pi*n)). - Vaclav Kotesovec, Jun 03 2015
a(n) = [x^n] 1/((1 - 2*x)*(1 - x)^(3*n)). - Ilya Gutkovskiy, Oct 25 2017
a(n) = Sum_{k=0..floor(n/2)} binomial(4*n+1,n-2*k). - Seiichi Manyama, Apr 09 2024
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+k-1,k). - Seiichi Manyama, Jul 30 2025
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k). - Seiichi Manyama, Aug 08 2025
G.f.: 1/(1 - x*g^3*(8-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 17 2025

A047099 a(n) = A047098(n)/2.

Original entry on oeis.org

1, 4, 19, 98, 531, 2974, 17060, 99658, 590563, 3540464, 21430267, 130771376, 803538100, 4967127736, 30866224824, 192696614730, 1207967820099, 7600482116932, 47981452358201, 303820299643138, 1929099000980219, 12279621792772822, 78346444891033856

Offset: 1

Clark Kimberling, Dec 11 1999

nonn

T(2*n,n)/2, with array T as in A047110.
Also given by a recurrence that features row 3 of the Pascal triangle (Mathematica code): u[0,0]=1; u[n_,k_]/;k<0 || k>n := 0; u[n_,k_]/;0<=k<=n := u[n,k] = u[n-1,k-1] + 3u[n-1,k] + 3u[n-1,k+1] + u[n-1,k+2]; u[n_]:=Sum[u[n,k],{k,0,n}]; Table[u[n],{n,0,10}]. - David Callan, Jul 22 2008
INVERT transform of (1,3,12,55,273,...), the ternary numbers A001764. - David Callan, Nov 21 2011

Alois P. Heinz, Table of n, a(n) for n = 1..400
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
J.-P. Bultel and S. Giraudo, Combinatorial Hopf algebras from PROs, arXiv preprint arXiv:1406.6903 [math.CO], 2014-2016.
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Noga Alon and Noah Kravitz, Counting Dope Matrices, arXiv:2205.09302 [math.CO], 2022.

Cf. A066380, A005809.
Column k=2 of A213027.
Cf. A001764.

f := n -> binomial(3*n, n) - (1/2)*add(binomial(3*n, k), k=0..n):
seq(f(n), n=1..20);

Table[Binomial[3 n, n] - Sum[Binomial[3 n, k], {k, 0, n}]/2, {n, 20}] (* Wesley Ivan Hurt, Jun 13 2014 *)

{a(n)=local(A=1+x); A=x*exp(sum(m=1, n+1, sum(j=0, m, binomial(3*m, j))*x^m/m +x*O(x^n))); polcoeff(A, n)} \\ Paul D. Hanna, Sep 04 2012

a(n) = binomial(3*n, n) - (1/2)*Sum_{k=0..n} binomial(3*n, k). - Vladeta Jovovic, Mar 22 2003
a(n) = A047098(n)/2. - Benoit Cloitre, Jan 28 2004
From Gary W. Adamson, Jul 28 2011: (Start)
a(n) is the upper left term in M^n, where M is the infinite square production matrix as follows:
  1, 1, 0, 0, 0, 0, ...
  3, 3, 1, 0, 0, 0, ...
  3, 3, 3, 1, 0, 0, ...
  1, 1, 3, 3, 1, 0, ...
  0, 0, 1, 3, 3, 0, ...
  0, 0, 0, 1, 3, 0, ...
  ... (End)
G.f.: x*exp( Sum_{n>=1} A066380*x^n/n ) where A066380(n) = Sum_{k=0..n} binomial(3*n,k). - Paul D. Hanna, Sep 04 2012
G.f.: (F(x)-1)/(2-F(x)), where F(x) is g.f. of A001764. - Vladimir Kruchinin, Jun 13 2014.
a(n) = (1/n)*Sum_{k=1..n} k*C(3*n,n-k). - Vladimir Kruchinin, Oct 03 2022
From Paul D. Hanna, Jun 06 2025: (Start)
G.f. A(x) = Series_Reversion( x*(1 + x)^2 / (1 + 2*x)^3 ).
G.f. satisfies A(x) = x*(1 + 2*A(x))^3 / (1 + A(x))^2.
G.f. satisfies A'(x) = A(x) * (1 + A(x)) * (1 + 2*A(x)) / (x*(1 - A(x))).
(End)

Comment revised by Clark Kimberling, Dec 08 2006

Edited by N. J. A. Sloane, Dec 21 2006

A371739 a(n) = Sum_{k=0..n} binomial(5*n,k).

Original entry on oeis.org

1, 6, 56, 576, 6196, 68406, 768212, 8731848, 100146724, 1156626990, 13432735556, 156713948672, 1835237017324, 21560768699762, 253994850228896, 2999267652451776, 35490014668470052, 420718526924212654, 4995548847105422048, 59402743684137281920

Offset: 0

Seiichi Manyama, Apr 05 2024

Cf. A032443, A066380, A066381.

Cf. A002294, A163455, A386699.

Table[32^n - Binomial[5*n, 1+n] * Hypergeometric2F1[1, 1 - 4*n, 2+n, -1], {n, 0, 20}] (* Vaclav Kotesovec, Apr 05 2024 *)

PARI
```
a(n) = sum(k=0, n, binomial(5*n, k));
```

a(n) = [x^n] 1/((1-2*x) * (1-x)^(4*n)).
a(n) ~ 5^(5*n + 1/2) / (3*sqrt(Pi*n) * 2^(8*n - 1/2)). - Vaclav Kotesovec, Apr 05 2024
a(n) = Sum_{k=0..floor(n/2)} binomial(5*n+1,n-2*k). - Seiichi Manyama, Apr 09 2024
a(n) = binomial(1+5*n, n)*hypergeom([1, (1-n)/2, -n/2], [1+2*n, 3/2+2*n], 1). - Stefano Spezia, Apr 09 2024
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+k-1,k). - Seiichi Manyama, Jul 30 2025
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n,k) * binomial(5*n-k-1,n-k). - Seiichi Manyama, Aug 08 2025
G.f.: g/((2-g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 12 2025
G.f.: 1/(1 - x*g^4*(10-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 17 2025

A387007 a(n) = Sum_{k=0..n} binomial(3*n+2,k).

Original entry on oeis.org

1, 6, 37, 232, 1471, 9402, 60460, 390656, 2533987, 16489546, 107594213, 703680424, 4611412196, 30273024984, 199045392232, 1310535994368, 8639411571051, 57017083602138, 376674527189599, 2490742704227192, 16483857933928471, 109175823528400778, 723611538997758784

Offset: 0

Seiichi Manyama, Aug 12 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A066380, A160906, A387008.

Cf. A001764.

[&+[Binomial(3*n+2,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025

Table[Sum[Binomial[3*n+2,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 27 2025 *)

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

a(n) = [x^n] (1+x)^(3*n+2)/(1-x).
a(n) = [x^n] 1/((1-x)^(2*n+2) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+2,k) * binomial(3*n-k+1,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k+1,n-k).
G.f.: g^3/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
D-finite with recurrence: 24*(5*n+1)*(3*n-1)*(3*n-2)*a(n-2) -(295*n^3-156*n^2-61*n+6)*a(n-1) +2*n*(2*n+1)*(5*n-4)*a(n). - Georg Fischer, Aug 17 2025

A387008 a(n) = Sum_{k=0..n} binomial(3*n+3,k).

Original entry on oeis.org

1, 7, 46, 299, 1941, 12616, 82160, 536155, 3505699, 22964087, 150676186, 990134948, 6515349244, 42925973608, 283134975936, 1869455684187, 12355133446527, 81725384344741, 541021064605298, 3584203906519219, 23761237400402597, 157623924396214756, 1046244086051121248

Offset: 0

Seiichi Manyama, Aug 12 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A066380, A160906, A387007.

Cf. A001764.

[&+[Binomial(3*n+3,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025

Table[Sum[Binomial[3*n+3,k], {k,0,n}], {n,0,25}] (* Vaclav Kotesovec, Aug 20 2025 *)

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

a(n) = [x^n] (1+x)^(3*n+3)/(1-x).
a(n) = [x^n] 1/((1-x)^(2*n+3) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+3,k) * binomial(3*n-k+2,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k+2,n-k).
G.f.: g^4/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
D-finite with recurrence: 24*(3*n-2)*(3*n-1)*(5*n^2+n-2)*a(n-2) -(295*n^4-156*n^3-339*n^2+12*n+20)*a(n-1) +2*(2*n+1)*(n+1)*(5*n^2-9*n+2)*a(n) = 0. - Georg Fischer, Aug 17 2025
a(n) ~ 3^(3*n + 7/2) / (sqrt(Pi*n) * 2^(2*n+3)). - Vaclav Kotesovec, Aug 20 2025

A371742 a(n) = Sum_{k=0..floor(n/2)} binomial(3n-k,n-2k).

Original entry on oeis.org

1, 3, 16, 92, 551, 3380, 21065, 132771, 843944, 5399802, 34731776, 224361283, 1454557294, 9458829681, 61670895633, 403003997300, 2638776935215, 17308508054848, 113709379928689, 748069400432262, 4927608724973776, 32495826854732633, 214521754579553129

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Cf. A108081, A371743, A371744.
Cf. A066380, A371754.
Cf. A183160.

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

a(n) = [x^n] 1/((1-x-x^2) * (1-x)^(2*n)).

a(n) ~ 3^(3*n + 3/2) / (5 * sqrt(Pi*n) * 2^(2*n)). - Vaclav Kotesovec, Apr 05 2024

A371754 a(n) = Sum_{k=0..floor(n/3)} binomial(3n-2k,n-3*k).

Original entry on oeis.org

1, 3, 15, 85, 505, 3081, 19125, 120173, 761995, 4865697, 31244029, 201544551, 1305039209, 8477521051, 55221311565, 360559717807, 2359123470971, 15463951609491, 101530816122729, 667587477393509, 4395294402200983, 28972295880583861, 191181607835416543

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Cf. A144904, A371755, A371756.

Cf. A066380, A371742.

Table[Sum[Binomial[3n-2k,n-3k],{k,0,Floor[n/3]}],{n,0,30}] (* Harvey P. Dale, Oct 19 2024 *)

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

a(n) = [x^n] 1/((1-x-x^3) * (1-x)^(2*n)).

a(n) ~ 3^(3*n + 5/2) / (17 * sqrt(Pi*n) * 2^(2*n)). - Vaclav Kotesovec, Apr 05 2024

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

A386700 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(3*n,k).

Original entry on oeis.org

1, 0, 6, 30, 186, 1140, 7116, 44856, 285066, 1823232, 11721726, 75683718, 490429224, 3187723344, 20774505408, 135699314640, 888177411018, 5823660624408, 38245666664994, 251528316024042, 1656338630258826, 10919849458481028, 72068276593960884, 476093333668519872

Offset: 0

Seiichi Manyama, Jul 30 2025

Cf. A122803, A386701, A386702.

Cf. A005809, A066380, A165817, A371813, A385004.

Table[(-8/9)^n - Binomial[3*n, n]*(-1 + Hypergeometric2F1[1, -2*n, 1 + n, 1/3]), {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)

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

a(n) = [x^n] 1/((1+2*x) * (1-x)^(2*n)).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(2*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 18*n*(2*n - 1)*(55*n^2 - 175*n + 138)*a(n) = (11605*n^4 - 49410*n^3 + 74243*n^2 - 46014*n + 9720)*a(n-1) + 24*(3*n - 5)*(3*n - 4)*(55*n^2 - 65*n + 18)*a(n-2).
a(n) ~ 3^(3*n + 1/2) / (5 * sqrt(Pi*n) * 2^(2*n)). (End)
G.f.: g/((-2+3*g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - 6*x*g^2*(-1+g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 17 2025

A387033 a(n) = Sum_{k=0..n} binomial(3*n-1,k).

Original entry on oeis.org

1, 3, 16, 93, 562, 3473, 21778, 137980, 880970, 5658537, 36519556, 236618693, 1538132224, 10026362492, 65513177704, 428957009288, 2813768603466, 18486790962201, 121634649321208, 801330506737399, 5285305708097522, 34896814868837161, 230631268849574378

Offset: 0

Seiichi Manyama, Aug 13 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A066380, A160906, A385823, A386006, A387007, A387008.
Cf. A114121, A387037.
Cf. A001764, A047099, A165817.

[&+[Binomial(3*n-1,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025

Table[Sum[Binomial[3*n-1,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 27 2025 *)

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

a(n) = [x^n] (1+x)^(3*n-1)/(1-x).
a(n) = [x^n] 1/((1-x)^(2*n-1) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n-1,k) * binomial(3*n-k-2,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k-2,n-k).
G.f.: 1/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
D-finite with recurrence: 24*(5*n+6)*(3*n-4)*(3*n-5)*a(n-2)-(295*n^3-451*n^2-234*n+360)*a(n-1)+2*n*(5*n+1)*(2*n-3)*a(n) = 0. - Georg Fischer, Aug 17 2025
a(n) ~ 3^(3*n - 1/2) / (sqrt(Pi*n) * 2^(2*n-1)). - Vaclav Kotesovec, Aug 27 2025

cp's OEIS Frontend

A066381 a(n) = Sum_{k=0..n} binomial(4*n,k).

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Maple

Mathematica

Maxima

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A047099 a(n) = A047098(n)/2.

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Maple

Mathematica

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A371739 a(n) = Sum_{k=0..n} binomial(5*n,k).

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Mathematica

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A387007 a(n) = Sum_{k=0..n} binomial(3*n+2,k).

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Magma

Mathematica

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A387008 a(n) = Sum_{k=0..n} binomial(3*n+3,k).

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Magma

Mathematica

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A371742 a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-k,n-2*k).

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A371754 a(n) = Sum_{k=0..floor(n/3)} binomial(3*n-2*k,n-3*k).

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Mathematica

PARI

Formula

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

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A371742 a(n) = Sum_{k=0..floor(n/2)} binomial(3n-k,n-2k).

A371754 a(n) = Sum_{k=0..floor(n/3)} binomial(3n-2k,n-3*k).