A066381 -id:A066381 - cp's OEIS frontend

A386811 a(n) = Sum_{k=0..n} binomial(4*n+1,k).

1, 6, 46, 378, 3214, 27896, 245506, 2182396, 19548046, 176142312, 1594831736, 14497410186, 132224930146, 1209397179048, 11088872706188, 101890087382168, 937973964234638, 8649109175873288, 79872298511230120, 738583466508887304, 6837944227813170424

Offset: 0

Seiichi Manyama, Aug 03 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A000302, A160906, A386812.
Cf. A005810, A078995, A147855, A226733, A226761, A385605.
Cf. A098430, A383716.
Cf. A066381.

[&+[Binomial(4*n+1, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 21 2025

Table[Sum[Binomial[4*n+1,k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *)

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

a(n) = [x^n] 1/((1-2*x) * (1-x)^(3*n+1)).
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+k,k).
D-finite with recurrence +645*n*(3*n-1)*(3*n-2)*a(n) +8*(-56722*n^3+213090*n^2-305978*n+150255)*a(n-1) +128*(62908*n^3-282348*n^2+385070*n-126735)*a(n-2) +12288*(-2486*n^3+8918*n^2+758*n-18935)*a(n-3) -2949120*(2*n-7)*(4*n-13)*(4*n-11)*a(n-4)=0. - R. J. Mathar, Aug 03 2025
a(n) = 2^(4*n+1) - binomial(4*n+1, n)*(hypergeom([1, -1-3*n], [1+n], -1) - 1). - Stefano Spezia, Aug 05 2025
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k). - Seiichi Manyama, Aug 07 2025
a(n) ~ 2^(8*n + 3/2) / (sqrt(Pi*n) * 3^(3*n + 1/2)). - Vaclav Kotesovec, Aug 07 2025
G.f.: g^2/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 12 2025
G.f.: B(x)^2/(1 + (B(x)-1)/2), where B(x) is the g.f. of A005810. - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^2*(8-2*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 16 2025

A066380 a(n) = Sum_{k=0..n} binomial(3*n,k).

Original entry on oeis.org

1, 4, 22, 130, 794, 4944, 31180, 198440, 1271626, 8192524, 53009102, 344212906, 2241812648, 14637774688, 95786202688, 628002401520, 4124304597834, 27126202533252, 178651732923346, 1178005033926998, 7776048412324714

Offset: 0

N. J. A. Sloane, Dec 23 2001

R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 425.

Stefano Spezia, Table of n, a(n) for n = 0..1200 (first 151 terms from Harry J. Smith)

Cf. A001764, A032443, A066381, A371739.

A066380:=n->add(binomial(3*n,k), k=0..n): seq(A066380(n), n=0..20); # Wesley Ivan Hurt, Sep 18 2014

Table[Sum[Binomial[3 n, k], {k, 0, n}], {n, 0, 20}] (* Geoffrey Critzer, May 27 2013 *)
a[n_] := 8^n - (2*n)/(n+1)*Binomial[3*n, n]*Hypergeometric2F1[1, -2*n+1, n+2, -1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 02 2013 *)

a[0]:1$ a[n]:=8*a[n-1]-(5*n^2+n-2)*(3*n-3)!/((2*n-1)!*n!)$ makelist(a[n],n,0,200); /* Tani Akinari, Sep 02 2014 */

{ for (n=0, 150, a=0; for (k=0, n, a+=binomial(3*n, k)); write("b066380.txt", n, " ", a) ) } \\ Harry J. Smith, Feb 12 2010

a(n) ~ C(3n, n)(2 - 4/n + O(1/n^2)).
G.f.: (1-g)/((3*g-1)*(2*g-1)) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011
G.f.: x*(d/dx)log((F(x)-1)/(2-F(x))), where F(x) is g.f. of A001764. - Vladimir Kruchinin, Jun 13 2014
a(0)=1, a(n) = 8*a(n-1) - (5*n^2+n-2)*(3*n-3)!/((2*n-1)!*n!). - Tani Akinari, Sep 02 2014
a(n) = [x^n] 1/((1 - 2*x)*(1 - x)^(2*n)). - Ilya Gutkovskiy, Oct 25 2017
a(n) = Sum_{k=0..floor(n/2)} binomial(3*n+1,n-2*k). - Seiichi Manyama, Apr 09 2024
a(n) = binomial(1+3*n, n)*hypergeom([1, (1-n)/2, -n/2], [1+n, 3/2+n], 1). - Stefano Spezia, Apr 09 2024
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+k-1,k). - Seiichi Manyama, Jul 30 2025
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k). - Seiichi Manyama, Aug 08 2025
From Seiichi Manyama, Aug 17 2025: (Start)
G.f.: 1/(1 - x*g^2*(6-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
G.f.: g/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. (End)

A387009 a(n) = Sum_{k=0..n} binomial(4*n+2,k).

Original entry on oeis.org

1, 7, 56, 470, 4048, 35443, 313912, 2804012, 25211936, 227881004, 2068564064, 18844224462, 172186125456, 1577401391626, 14483100716176, 133240186921816, 1227901991526976, 11333497984085620, 104752914242685856, 969417048912326008, 8981452266787224128

Offset: 0

Seiichi Manyama, Aug 12 2025

nonn

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

Cf. A066381, A386811, A387010, A387011.

Cf. A002293.

[&+[Binomial(4*n+2, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 16 2025

Table[Sum[Binomial[4*n+2,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 16 2025 *)

a(n) = sum(k=0, n, binomial(4*n+2, k));

a(n) = [x^n] (1+x)^(4*n+2)/(1-x).
a(n) = [x^n] 1/((1-x)^(3*n+2) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+2,k) * binomial(4*n-k+1,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k+1,n-k).
G.f.: g^3/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
D-finite with recurrence: 128*(4*n-3)*(2*n-1)*(4*n-5)*(22*n+5)*a(n-2) -8*(1892*n^4-3706*n^3+1750*n^2+214*n-177)*a(n-1) +3*(22*n-17)*(n-1)*(3*n-1)*(3*n+1)*a(n) = 0. - Georg Fischer, Aug 17 2025

A387010 a(n) = Sum_{k=0..n} binomial(4*n+3,k).

Original entry on oeis.org

1, 8, 67, 576, 5036, 44552, 397594, 3572224, 32267668, 292750368, 2665685155, 24347665728, 222972599812, 2046626681072, 18823260696452, 173427623923712, 1600383346290116, 14789063407109600, 136838247669241276, 1267571539176770816, 11754134090271100336

Offset: 0

Seiichi Manyama, Aug 12 2025

nonn

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

Cf. A066381, A386811, A387009, A387011.

Cf. A002293.

[&+[Binomial(4*n+3, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 16 2025

Table[Sum[Binomial[4*n+3,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 16 2025 *)

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

a(n) = [x^n] (1+x)^(4*n+3)/(1-x).
a(n) = [x^n] 1/((1-x)^(3*n+3) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+3,k) * binomial(4*n-k+2,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k+2,n-k).
G.f.: g^4/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
D-finite with recurrence: 128*(4*n-1)*(2*n-1)*(4*n-3)*(11*n^2+8*n+1)*a(n-2) -8*(946*n^5-434*n^4-518*n^3+143*n^2+46*n-3)*a(n-1) +3*n*(3*n+1)*(3*n+2)*(11*n^2-14*n+4)*a(n) = 0. - Georg Fischer, Aug 17 2025

A387011 a(n) = Sum_{k=0..n} binomial(4*n+4,k).

Original entry on oeis.org

1, 9, 79, 697, 6196, 55455, 499178, 4514873, 40999516, 373585604, 3414035527, 31278197839, 287191809724, 2642070371194, 24347999094724, 224723513577529, 2076978797223820, 19220104372823340, 178061257422521452, 1651314042800498052, 15328459501269535952

Offset: 0

Seiichi Manyama, Aug 12 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A066381, A386811, A387009, A387010.

Cf. A002293.

[&+[Binomial(4*n+4, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 16 2025

Table[Sum[Binomial[4*n+4,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 16 2025 *)

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

a(n) = [x^n] (1+x)^(4*n+4)/(1-x).
a(n) = [x^n] 1/((1-x)^(3*n+4) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+4,k) * binomial(4*n-k+3,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k+3,n-k).
G.f.: g^5/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
D-finite with recurrence: 128*(4*n-3)*(2*n+1)*(4*n-1)*(22*n^2+16*n-3)*a(n-2) -8*(1892*n^5+1024*n^4-1982*n^3-1306*n^2-60*n+27)*a(n-1) +3*(n+1)*(3*n+2)*(3*n+1)*(22*n^2-28*n+3)*a(n) = 0. - Georg Fischer, Aug 17 2025
a(n) ~ 2^(8*n + 15/2) / (sqrt(Pi*n) * 3^(3*n + 7/2)). - Vaclav Kotesovec, Aug 18 2025

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

A385498 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n,k).

Original entry on oeis.org

1, 6, 48, 408, 3564, 31626, 283548, 2560872, 23255964, 212101176, 1941110628, 17815257048, 163896843300, 1510891524252, 13952756564424, 129048895061208, 1195191116753436, 11082661017288264, 102877353868090080, 955912961224763232, 8889969049985302464

Offset: 0

Seiichi Manyama, Jul 30 2025

Cf. A100192, A385004, A386699.

Cf. A005810, A066381, A371814, A386701.

Table[(81/8)^n - Binomial[4*n, n]*(-1 + Hypergeometric2F1[1, -3*n, 1 + n, -1/2]), {n,0,25}] (* Vaclav Kotesovec, Jul 30 2025 *)

a(n) = sum(k=0, n, 2^(n-k)*binomial(4*n, k));

a(n) = [x^n] 1/((1-3*x) * (1-x)^(3*n)).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(3*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 24*n*(3*n - 2)*(3*n - 1)*(139*n^3 - 366*n^2 + 143*n + 132)*a(n) = (588665*n^6 - 2281011*n^5 + 2262209*n^4 + 1245939*n^3 - 3359986*n^2 + 1877400*n - 322560)*a(n-1) - 648*(2*n - 3)*(4*n - 7)*(4*n - 5)*(139*n^3 + 51*n^2 - 172*n + 48)*a(n-2).
a(n) ~ 2^(8*n + 1/2) / (sqrt(Pi*n) * 3^(3*n - 1/2)). (End)
G.f.: g/((3-2*g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^3*(12-6*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 17 2025

A386701 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(4*n,k).

Original entry on oeis.org

1, 1, 13, 103, 869, 7476, 65323, 577242, 5144949, 46167196, 416527828, 3774785983, 34336862435, 313330665532, 2866982877954, 26294890918308, 241665561294741, 2225104901535564, 20520648006149980, 189523353219338572, 1752680220372189364, 16227703263403842768

Offset: 0

Seiichi Manyama, Jul 30 2025

Cf. A386700, A386702.

Cf. A066381, A262977, A371814, A385498.

Table[(-16/27)^n - Binomial[4*n, n]*(-1 + Hypergeometric2F1[1, -3*n, 1 + n, 1/3]), {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)

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

a(n) = [x^n] 1/((1+2*x) * (1-x)^(3*n)).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 81*n*(3*n - 2)*(3*n - 1)*(612*n^3 - 2838*n^2 + 4354*n - 2209)*a(n) = 24*(165240*n^6 - 1019628*n^5 + 2493432*n^4 - 3068178*n^3 + 1984652*n^2 - 632900*n + 76545)*a(n-1) + 128*(2*n - 3)*(4*n - 7)*(4*n - 5)*(612*n^3 - 1002*n^2 + 514*n - 81)*a(n-2).
a(n) ~ 2^(8*n - 1/2) / (sqrt(Pi*n) * 3^(3*n + 1/2)). (End)
G.f.: g/((-2+3*g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^3*(-8+9*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 17 2025

A387037 a(n) = Sum_{k=0..n} binomial(4*n-1,k).

Original entry on oeis.org

1, 4, 29, 232, 1941, 16664, 145499, 1285624, 11460949, 102875128, 928495764, 8417689504, 76599066579, 699232769512, 6400175653922, 58718827590992, 539822826733397, 4971747032359352, 45863130731297180, 423683961417124576, 3919058645835901556

Offset: 0

Seiichi Manyama, Aug 13 2025

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

Cf. A066381, A386811, A387009, A387010, A387011, A387034, A387035, A387036.
Cf. A114121, A387033.
Cf. A002293, A262977.

[&+[Binomial(4*n-1, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025

Table[Sum[Binomial[4*n-1,k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *)

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

a(n) = [x^n] (1+x)^(4*n-1)/(1-x).
a(n) = [x^n] 1/((1-x)^(3*n-1) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n-1,k) * binomial(4*n-k-2,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k-2,n-k).
G.f.: 1/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
D-finite with recurrence: 128*(4*n-5)*(4*n-7)*(2*n-3)*(11*n^2-3*n-3)*a(n-2) -8*(946*n^5-4218*n^4+6512*n^3-3753*n^2+201*n+315)*a(n-1) +3*n*(3*n-2)*(3*n-4)*(11*n^2-25*n+11)*a(n) = 0. - Georg Fischer, Aug 17 2025
a(n) ~ 2^(8*n - 5/2) / (sqrt(Pi*n) * 3^(3*n - 3/2)). - Vaclav Kotesovec, Sep 03 2025

A371779 a(n) = Sum_{k=0..floor(n/3)} binomial(4n+2,n-3k).

Original entry on oeis.org

1, 6, 45, 365, 3078, 26565, 232831, 2063235, 18435021, 165780758, 1498533273, 13603087800, 123920995101, 1132284232215, 10372554403620, 95233251146671, 876081280823430, 8073359613286509, 74513645742072841, 688682977876117698, 6373025238727622277

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Cf. A371777, A371778, A371780.

Cf. A066381.

a(n) = sum(k=0, n\3, binomial(4*n+2, n-3*k));

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

cp's OEIS Frontend

A386811 a(n) = Sum_{k=0..n} binomial(4*n+1,k).

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A066380 a(n) = Sum_{k=0..n} binomial(3*n,k).

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A387009 a(n) = Sum_{k=0..n} binomial(4*n+2,k).

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A387010 a(n) = Sum_{k=0..n} binomial(4*n+3,k).

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A387011 a(n) = Sum_{k=0..n} binomial(4*n+4,k).

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Magma

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

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A385498 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n,k).

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Mathematica

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A386701 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(4*n,k).

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A371779 a(n) = Sum_{k=0..floor(n/3)} binomial(4n+2,n-3k).