A386811 -id:A386811 - cp's OEIS frontend

A078995 a(n) = Sum_{k=0..n} C(4k,k)C(4*(n-k),n-k).

1, 8, 72, 664, 6184, 57888, 543544, 5113872, 48180456, 454396000, 4288773152, 40503496536, 382701222296, 3617396099936, 34203591636048, 323492394385824, 3060238763412072, 28955508198895584, 274018698082833760, 2593539713410178528, 24550565251665845664

Offset: 0

N. J. A. Sloane, Jan 19 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Y_n for s=4).
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.

See A049235 for more information.
Cf. A006256, A079563, A079678, A079679.
Cf. A005810, A147855, A226733, A226761, A385605, A386811.

series(eval(g/(3*g-4), g=RootOf(g = 1+x*g^4,g))^2, x=0, 30); # Mark van Hoeij, May 06 2013

Table[Sum[Binomial[4*k, k]*Binomial[4*(n - k), n - k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 06 2012 *)

a(n) = sum(k=0, n, binomial(4*k, k)*binomial(4*(n-k), n-k)); \\ Michel Marcus, May 09 2020

a(n) = 2/3*(256/27)^n*(1+c/sqrt(n)+o(n^-1/2)) where c = 2/3*sqrt(2/(3*Pi)) = 0.307105910641187... More generally, 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. See A000302, A006256 for cases m=2 and 3. - Benoit Cloitre, Jan 26 2003, corrected and extended by Vaclav Kotesovec, Nov 06 2012
243*n*(8*n - 17)*(3*n - 1)*(3*n - 4)*(3*n - 2)*(3*n - 5)*a(n) = 72*(3*n - 5)*(3*n - 4)*(6912*n^4 - 33120*n^3 + 58256*n^2 - 47798*n + 15309)*a(n - 1) - 3072*(2*n - 3)*(6912*n^5 - 55008*n^4 + 175696*n^3 - 282180*n^2 + 227825*n - 73710)*a(n - 2) + 262144*(n - 2)*(4*n - 7)*(2*n - 3)*(2*n - 5)*(4*n - 9)*(8*n - 9)*a(n - 3). - Vladeta Jovovic, Jul 16 2004
Shorter recurrence: 81*n*(3*n-2)*(3*n-1)*(8*n-11)*a(n) = 24*(4608*n^4-14400*n^3+15776*n^2-7346*n+1215)*a(n-1) - 2048*(2*n-3)*(4*n-5)*(4*n-3)*(8*n-3)*a(n-2). - Vaclav Kotesovec, Nov 06 2012
a(n) = Sum_{k=0..n} binomial(4*k+l,k) * binomial(4*(n-k)-l,n-k) for every real number l. - Rui Duarte and António Guedes de Oliveira, Feb 16 2013
From Rui Duarte and António Guedes de Oliveira, Feb 17 2013: (Start)
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} 4^(n-k) * binomial(3*n+k,k). (End)
G.f.: g^2/(3*g-4)^2 where g=ogf(A002293) satisfies g = 1+x*g^4. - Mark van Hoeij, May 06 2013
a(n) = [x^n] 1/((1-4*x) * (1-x)^(3*n+1)). - Seiichi Manyama, Aug 03 2025
a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k). - Seiichi Manyama, Aug 15 2025

A160906 Row sums of A159841.

Original entry on oeis.org

1, 5, 29, 176, 1093, 6885, 43796, 280600, 1807781, 11698223, 75973189, 494889092, 3231947596, 21153123932, 138712176296, 911137377456, 5993760282021, 39481335979779, 260377117268087, 1719026098532296, 11360252318843933, 75141910203168229, 497431016774189912

Offset: 0

R. J. Mathar, May 29 2009

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

Cf. A075273, A159841.
Cf. A005809, A006256, A183160, A226751.
Cf. A000302, A386811, A386812.
Cf. A244038, A383326.

A160906 := proc(n) add( A159841(n,k), k=0..n) ; end:
seq(A160906(n), n=0..20) ;

Table[Sum[Binomial[3*n+1, 2*n+k+1], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 25 2017 *)

a(n) = sum(k=0, n, binomial(3*n+1, 2*n+k+1)); \\ Michel Marcus, Oct 31 2017

a = lambda n: binomial(3*n+1,n)*hypergeometric([1,-n],[2*n+2],-1)
[simplify(a(n)) for n in range(21)] # Peter Luschny, May 19 2015

a(n) = Sum_{k=0..n} A159841(n,k).
Conjecture: a(2n+1) = A075273(3n).
a(n) = C(3*n+1,n)*Hyper2F1([1,-n],[2*n+2],-1). - Peter Luschny, May 19 2015
Conjecture: 2*n*(2*n-1)*(5*n-4)*a(n) +(-295*n^3+451*n^2-130*n-24)*a(n-1) +24*(5*n+1)*(3*n-4)*(3*n-2)*a(n-2) = 0. - R. J. Mathar, Jul 20 2016
a(n) = [x^n] 1/((1 - 2*x)*(1 - x)^(2*n+1)). - Ilya Gutkovskiy, Oct 25 2017
a(n) ~ 3^(3*n + 3/2) / (sqrt(Pi*n) * 2^(2*n + 1)). - Vaclav Kotesovec, Oct 25 2017
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+k,k). - Seiichi Manyama, Aug 03 2025
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k). - Seiichi Manyama, Aug 07 2025
G.f.: g^2/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 12 2025
G.f.: B(x)^2/(1 + (B(x)-1)/3), where B(x) is the g.f. of A005809. - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g*(6-g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 16 2025

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

A147855 G.f.: 1 / (1 + 4xG(x)^2 - 7xG(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 3, 22, 174, 1444, 12323, 107104, 942952, 8381596, 75053100, 676017962, 6118171326, 55591175956, 506805088026, 4633571685968, 42468065811884, 390071875757852, 3589637747968964, 33089300640166360, 305476314574338648, 2823932709938708824, 26137341654281261347

Offset: 0

Paul D. Hanna, Jun 16 2013

nonn

			G.f.: A(x) = 1 + 3*x + 22*x^2 + 174*x^3 + 1444*x^4 + 12323*x^5 +...
A related series is G(x) = 1 + x*G(x)^4, where
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 + 17710*x^6 +...
G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 + 32890*x^6 +...
such that A(x) = 1/(1 + 4*x*G(x)^2 - 7*x*G(x)^3).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A226705, A002293.
Cf. A026641, A183160, A371753.
Cf. A005810, A078995, A226733, A226761, A385605, A386811.

Table[Sum[Binomial[2*n+k,n-k]*Binomial[2*n-k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 16 2013 *)

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

{a(n)=sum(k=0, n, binomial(k, n-k)*binomial(4*n-k, k))}
for(n=0, 30, print1(a(n), ", "))

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

{a(n)=local(G=1+x); for(i=0, n,G=1+x*G^4+x*O(x^n)); polcoeff(1/(1-3*x*G^2-7*x^2*G^6), n)}
for(n=0, 30, print1(a(n), ", "))

a(n) = Sum_{k=0..n} C(k, n-k) * C(4*n-k, k).
a(n) = Sum_{k=0..n} C(n+k, n-k) * C(3*n-k, k).
a(n) = Sum_{k=0..n} C(2*n+k, n-k) * C(2*n-k, k).
a(n) = Sum_{k=0..n} C(3*n+k, n-k) * C(n-k, k).
a(n) = Sum_{k=0..n} C(4*n+k, n-k) * C(-k, k).
G.f.: 1 / (1 - 3*x*G(x)^2 - 7*x^2*G(x)^6) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
a(n) ~ 2^(8*n+5/2)/(5*sqrt(Pi*n)*3^(3*n+1/2)). - Vaclav Kotesovec, Jun 16 2013
From Seiichi Manyama, Apr 05 2024: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(4*n-2*k-1,n-2*k).
a(n) = [x^n] 1/((1-x^2) * (1-x)^(3*n)). (End)
From Seiichi Manyama, Aug 05 2025: (Start)
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(3*n+k,k). (End)
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).
G.f.: G(x)^2/((-1+2*G(x)) * (4-3*G(x))) where G(x) = 1+x*G(x)^4 is the g.f. of A002293. (End)
G.f.: B(x)^2/(1 + 5*(B(x)-1)/4), where B(x) is the g.f. of A005810. - Seiichi Manyama, Aug 15 2025

A226733 G.f.: 1 / (1 + 8xG(x)^2 - 10xG(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 2, 18, 142, 1186, 10152, 88414, 779508, 6936066, 62159224, 560238728, 5072970366, 46114086446, 420558296888, 3846232573236, 35261290343112, 323952686556354, 2981787128165592, 27491128592627800, 253835886034173848, 2346892194318851016, 21724880414632781472

Offset: 0

Paul D. Hanna, Jun 16 2013

nonn

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 142*x^3 + 1186*x^4 + 10152*x^5 +...
A related series is G(x) = 1 + x*G(x)^4, where
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 + 17710*x^6 +...
G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 + 32890*x^6 +...
such that A(x) = 1/(1 + 8*x*G(x)^2 - 10*x*G(x)^3).

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

Cf. A147855, A183160, A226705, A002293.

Cf. A005810, A078995, A147855, A226761, A385605, A386811.

Table[Sum[Binomial[2*n+2*k,n-k]*Binomial[2*n-2*k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 16 2013 *)

{a(n)=local(G=1+x); for(i=0, n,G=1+x*G^4+x*O(x^n)); polcoeff(1/(1+8*x*G^2-10*x*G^3), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=local(G=1+x); for(i=0, n,G=1+x*G^4+x*O(x^n)); polcoeff(1/(1-2*x*G^2-10*x^2*G^6), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=sum(k=0, n, binomial(2*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(4*n-2*k, k))}
for(n=0, 30, print1(a(n), ", "))

{a(n)=sum(k=0, n, binomial(4*n+2*k, n-k)*binomial(-2*k, k))}
for(n=0, 30, print1(a(n), ", "))

a(n) = Sum_{k=0..n} C(2*k, n-k) * C(4*n-2*k, k).
a(n) = Sum_{k=0..n} C(n+2*k, n-k) * C(3*n-2*k, k).
a(n) = Sum_{k=0..n} C(2*n+2*k, n-k) * C(2*n-2*k, k).
a(n) = Sum_{k=0..n} C(3*n+2*k, n-k) * C(n-2*k, k).
a(n) = Sum_{k=0..n} C(4*n+2*k, n-k) * C(-2*k, k).
G.f.: 1 / (1 - 2*x*G(x)^2 - 10*x^2*G(x)^6) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
a(n) ~ 2^(8*n+3/2)/(3^(3*n+3/2)*sqrt(Pi*n)). - Vaclav Kotesovec, Jun 16 2013
From Seiichi Manyama, Aug 05 2025: (Start)
a(n) = [x^n] 1/((1+2*x) * (1-x)^(3*n+1)).
a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n+k,k). (End)
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).
G.f.: G(x)^2/((-2+3*G(x)) * (4-3*G(x))) where G(x) = 1+x*G(x)^4 is the g.f. of A002293. (End)
G.f.: B(x)^2/(1 + 3*(B(x)-1)/2), where B(x) is the g.f. of A005810. - Seiichi Manyama, Aug 15 2025

A385605 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+1,k).

Original entry on oeis.org

1, 7, 58, 502, 4436, 39687, 358024, 3249288, 29624796, 271080124, 2487835678, 22888216006, 211010997716, 1948830506578, 18026768864736, 166976297995452, 1548523206590364, 14376415735219572, 133599985919343400, 1242638966005222648, 11567295503871866536

Offset: 0

Seiichi Manyama, Aug 03 2025

nonn

Cf. A006256, A141223, A385632.
Cf. A005810, A078995, A147855, A386811.
Cf. A385498.

a(n) = sum(k=0, n, 2^(n-k)*binomial(4*n+1, k));

a(n) = [x^n] 1/((1-3*x) * (1-x)^(3*n+1)).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(3*n+k,k).
a(n) = 3^(4*n+1)*2^(-3*n-1) - binomial(4*n+1, n)*(hypergeom([1, -1-3*n], [1+n], -1/2) - 1). - Stefano Spezia, Aug 05 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k). - Seiichi Manyama, Aug 07 2025
G.f.: g^2/((3-2*g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 14 2025
G.f.: B(x)^2/(1 + (B(x)-1)/4), where B(x) is the g.f. of A005810. - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^2*(12-5*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 16 2025

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

A226761 G.f.: 1 / (1 + 12xG(x)^2 - 13xG(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 16, 118, 1004, 8601, 75076, 662796, 5903676, 52949332, 477533356, 4326309406, 39343725716, 358943047438, 3283745710968, 30112624408488, 276715616909148, 2547523969430508, 23491659440021920, 216942761366305144, 2006084011596742384, 18572529488934397689

Offset: 0

Paul D. Hanna, Jun 16 2013

nonn

			G.f.: A(x) = 1 + x + 16*x^2 + 118*x^3 + 1004*x^4 + 8601*x^5 +...
A related series is G(x) = 1 + x*G(x)^4, where
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 + 17710*x^6 +...
G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 + 32890*x^6 +...
such that A(x) = 1/(1 + 12*x*G(x)^2 - 13*x*G(x)^3).

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

Cf. A002293.

Cf. A005810, A078995, A147855, A226733, A385605, A386811.

Table[Sum[Binomial[2*n+3*k,n-k]*Binomial[2*n-3*k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 17 2013 *)

{a(n)=local(G=1+x); for(i=0, n, G=1+x*G^4+x*O(x^n)); polcoeff(1/(1+12*x*G^2-13*x*G^3), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=local(G=1+x); for(i=0, n, G=1+x*G^4+x*O(x^n)); polcoeff(1/(1-x*G^2-13*x^2*G^6), n)}
for(n=0, 30, print1(a(n), ", "))

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

{a(n)=sum(k=0, n, binomial(3*k, n-k)*binomial(4*n-3*k, k))}
for(n=0, 30, print1(a(n), ", "))

{a(n)=sum(k=0, n, binomial(4*n+3*k, n-k)*binomial(-3*k, k))}
for(n=0, 30, print1(a(n), ", "))

a(n) = Sum_{k=0..n} C(3*k, n-k) * C(4*n-3*k, k).
a(n) = Sum_{k=0..n} C(n+3*k, n-k) * C(3*n-3*k, k).
a(n) = Sum_{k=0..n} C(2*n+3*k, n-k) * C(2*n-3*k, k).
a(n) = Sum_{k=0..n} C(3*n+3*k, n-k) * C(n-3*k, k).
a(n) = Sum_{k=0..n} C(4*n+3*k, n-k) * C(-3*k, k).
G.f.: 1 / (1 - x*G(x)^2 - 13*x^2*G(x)^6) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
a(n) ~ 2^(8*n+5/2)/(7*3^(3*n+1/2)*sqrt(Pi*n)). - Vaclav Kotesovec, Jun 17 2013
From Seiichi Manyama, Aug 05 2025: (Start)
a(n) = [x^n] 1/((1+3*x) * (1-x)^(3*n+1)).
a(n) = Sum_{k=0..n} (-4)^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(3*n+k,k). (End)
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-3)^k * 4^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).
G.f.: G(x)^2/((-3+4*G(x)) * (4-3*G(x))) where G(x) = 1+x*G(x)^4 is the g.f. of A002293. (End)
G.f.: B(x)^2/(1 + 7*(B(x)-1)/4), where B(x) is the g.f. of A005810. - Seiichi Manyama, Aug 15 2025

cp's OEIS Frontend

A078995 a(n) = Sum_{k=0..n} C(4*k,k)*C(4*(n-k),n-k).

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A160906 Row sums of A159841.

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

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A147855 G.f.: 1 / (1 + 4*x*G(x)^2 - 7*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

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A226733 G.f.: 1 / (1 + 8*x*G(x)^2 - 10*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

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A385605 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+1,k).

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

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A078995 a(n) = Sum_{k=0..n} C(4k,k)C(4*(n-k),n-k).

A147855 G.f.: 1 / (1 + 4xG(x)^2 - 7xG(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

A226733 G.f.: 1 / (1 + 8xG(x)^2 - 10xG(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

A226761 G.f.: 1 / (1 + 12xG(x)^2 - 13xG(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.