A371739 -id:A371739 - 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

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)

A386812 a(n) = Sum_{k=0..n} binomial(5*n+1,k).

Original entry on oeis.org

1, 7, 67, 697, 7547, 83682, 942649, 10739176, 123388763, 1427090845, 16593192942, 193774331494, 2271115189673, 26700463884244, 314735943548632, 3718522618187472, 44021808206431579, 522080025971331983, 6201449551502245321, 73767447652621434695, 878599223738760686422

Offset: 0

Seiichi Manyama, Aug 03 2025

nonn

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

Cf. A000302, A160906, A386811.
Cf. A002294, A371739.
Cf. A001449, A079589, A079678, A371753, A385632.

[&+[Binomial(5*n+1, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 21 2025

Table[Sum[Binomial[5*n+1,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 21 2025 *)

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

a(n) = [x^n] 1/((1-2*x) * (1-x)^(4*n+1)).
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+k,k).
D-finite with recurrence 8*n*(2754528070303487*n -4672004545621835)*(4*n-3)*(2*n-1) *(4*n-1)*a(n) +(-5828620079131711179*n^5 -135826272187971586019*n^4 +779361612339655552281*n^3 -1570139520911413863589*n^2 +1419656431480813021170*n -487668485184225269400)*a(n-1) +40*(-21123668262204329085*n^5 +243394620512022153401*n^4 -982249084763267479011*n^3 +1849334401749026834935*n^2 -1662134287466221884960*n +573649997457991096080)*a(n-2) +6400*(5*n-13)*(5*n-11)*(2475036532470005*n-2376524337096748)*(5*n-9)*(5*n-12)*a(n-3)=0. - R. J. Mathar, Aug 03 2025
a(n) = 2^(5*n+1) - binomial(5*n+1, n)*(hypergeom([1, -1-4*n], [1+n], -1) - 1). - Stefano Spezia, Aug 05 2025
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k) * binomial(5*n-k,n-k). - Seiichi Manyama, Aug 07 2025
G.f.: g^2/((2-g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 12 2025
From Seiichi Manyama, Aug 16 2025: (Start)
G.f.: 1/(1 - x*g^3*(10-3*g)) where g = 1+x*g^5 is the g.f. of A002294.
G.f.: B(x)^2/(1 + 3*(B(x)-1)/5), where B(x) is the g.f. of A001449. (End)
a(n) ~ 5^(5*n + 3/2) / (3*sqrt(Pi*n) * 2^(8*n + 3/2)). - Vaclav Kotesovec, Aug 21 2025

A386699 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(5*n,k).

Original entry on oeis.org

1, 7, 69, 733, 8061, 90462, 1028871, 11814376, 136643085, 1589311381, 18569375114, 217773347502, 2561944357311, 30219704365104, 357278540928168, 4232449819704768, 50227362114232109, 596988743410929087, 7105534815529752831, 84678089652554263155, 1010268312800732117946

Offset: 0

Seiichi Manyama, Jul 30 2025

Cf. A000244, A100192, A385004, A385498.
Cf. A163455, A371739, A386702.
Cf. A002294.

Table[(243/16)^n - Binomial[5*n, n]*(-1 + Hypergeometric2F1[1, -4*n, 1 + n, -1/2]), {n,0,25}] (* Vaclav Kotesovec, Jul 30 2025 *)

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

a(n) = [x^n] 1/((1-3*x) * (1-x)^(4*n)).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(4*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 128*n*(2*n - 1)*(4*n - 3)*(4*n - 1)*(3052*n^4 - 15114*n^3 + 26432*n^2 - 18693*n + 4131)*a(n) = 8*(42807352*n^8 - 285737492*n^7 + 758983420*n^6 - 1002945218*n^5 + 644348866*n^4 - 111879380*n^3 - 84004497*n^2 + 44187381*n - 5806080)*a(n-1) - 1215*(5*n - 9)*(5*n - 8)*(5*n - 7)*(5*n - 6)*(3052*n^4 - 2906*n^3 - 598*n^2 + 1037*n - 192)*a(n-2).
a(n) ~ 5^(5*n + 1/2) / (sqrt(Pi*n) * 2^(8*n + 1/2)). (End)
G.f.: g/((3-2*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(5*n,k) * binomial(5*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^4*(15-8*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 17 2025

A371780 a(n) = Sum_{k=0..floor(n/3)} binomial(5n+2,n-3k).

Original entry on oeis.org

1, 7, 66, 681, 7337, 81081, 911153, 10361554, 118881714, 1373402934, 15954079557, 186165866937, 2180501226751, 25620628577083, 301858589475117, 3564841627421691, 42186363329210473, 500142626996777355, 5939062937833796486, 70626949319708756435

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Robert Israel, Table of n, a(n) for n = 0..918

Cf. A371777, A371778, A371779.

Cf. A371739.

f:= proc(n) local k; add(binomial(5*n+2,n-3*k),k=0..n/3); end proc:
map(f, [$0..100]); # Robert Israel, Apr 22 2024

a(n) = sum(k=0, n\3, binomial(5*n+2, n-3*k));

a(n) = [x^n] 1/(((1-x)^3-x^3) * (1-x)^(4*n)).

A386702 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(5*n,k).

Original entry on oeis.org

1, 2, 24, 248, 2676, 29562, 331956, 3771896, 43242660, 499215146, 5795429764, 67587697872, 791232339756, 9292673328174, 109440405341088, 1291977861163968, 15284200451058724, 181147979395807002, 2150493166839159936, 25567085678133719880, 304368033788893315896

Offset: 0

Seiichi Manyama, Jul 30 2025

Cf. A386700, A386701.

Cf. A163455, A371739, A386699.

Table[(-32/81)^n - Binomial[5*n, n]*(-1 + Hypergeometric2F1[1, -4*n, 1 + n, 1/3]), {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)

a(n) = sum(k=0, n, (-3)^(n-k)*binomial(5*n, k));

a(n) = [x^n] 1/((1+2*x) * (1-x)^(4*n)).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(4*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 648*n*(2*n - 1)*(4*n - 3)*(4*n - 1)*(37331*n^4 - 227972*n^3 + 518701*n^2 - 521044*n + 194928)*a(n) = (9143593823*n^8 - 74277961298*n^7 + 253684378280*n^6 - 473415527402*n^5 + 524935655069*n^4 - 351762123620*n^3 + 137998180332*n^2 - 28677229776*n + 2380855680)*a(n-1) + 160*(5*n - 9)*(5*n - 8)*(5*n - 7)*(5*n - 6)*(37331*n^4 - 78648*n^3 + 58771*n^2 - 18234*n + 1944)*a(n-2).
a(n) ~ 5^(5*n + 1/2) / (7 * sqrt(Pi*n) * 2^(8*n - 1/2)). (End)
G.f.: g/((-2+3*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(5*n,k) * binomial(5*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^4*(-10+12*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 17 2025

cp's OEIS Frontend

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

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

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A386812 a(n) = Sum_{k=0..n} binomial(5*n+1,k).

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Magma

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A386699 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(5*n,k).

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A371780 a(n) = Sum_{k=0..floor(n/3)} binomial(5*n+2,n-3*k).

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Maple

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Formula

A386702 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(5*n,k).

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Mathematica

PARI

Formula

A371780 a(n) = Sum_{k=0..floor(n/3)} binomial(5n+2,n-3k).