A386812 -id:A386812 - 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

A079678 a(n) = a(n,m) = Sum_{k=0..n} binomial(mk,k)binomial(m*(n-k),n-k) for m=5.

Original entry on oeis.org

1, 10, 115, 1360, 16265, 195660, 2361925, 28577440, 346316645, 4201744870, 51023399190, 620022989200, 7538489480075, 91696845873760, 1115794688036920, 13581508654978560, 165357977228808925, 2013721466517360650, 24527742112263770425, 298805688708113438240, 3640695209795092874290

Offset: 0

Benoit Cloitre, Jan 26 2003

nonn

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, A078995 for cases m=2,3 and 4.

Robert Israel, Table of n, a(n) for n = 0..828
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344.
Rui Duarte and António Guedes de Oliveira, Short note on the convolution of binomial coefficients, arXiv:1302.2100 [math.CO], 2013.

Cf. A006256, A078995, A079563, A079679.

Cf. A371753, A385632, A386812.

seq(add(binomial(5*k,k)*binomial(5*(n-k),n-k),k=0..n), n=0..30); # Robert Israel, Jul 16 2015

m = 5; Table[Sum[Binomial[m k, k] Binomial[m (n - k), n - k], {k, 0, n}], {n, 0, 17}] (* Michael De Vlieger, Sep 30 2015 *)

main(size)=my(k,n,m=5); concat(1,vector(size,n, sum(k=0,n, binomial(m*k,k)*binomial(m*(n-k),n-k)))) \\ Anders Hellström, Jul 16 2015

a(n) = sum(k=0,n,4^(n-k)*binomial(5*n+1,k));
vector(30, n, a(n-1)) \\ Altug Alkan, Sep 30 2015

a(n) = 5/8*(3125/256)^n*(1+c/sqrt(n)+o(n^-1/2)) where c=0.356...
c = sqrt(2)/sqrt(5*Pi) = 0.3568248232305542229... - Vaclav Kotesovec, May 25 2020
a(n) = Sum_{k=0..n} binomial(5*k+l,k) * binomial(5*(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} 4^(n-k) * binomial(5*n+1,k).
a(n) = Sum_{k=0..n} 5^(n-k) * binomial(4*n+k,k). (End)
G.f.: hypergeom([1/5, 2/5, 3/5, 4/5], [1/4, 1/2, 3/4], (3125/256)*x)^2 satisfies
((3125/2)*g^3*x^4-128*g^3*x^3)*g''''+((-3125*g^2*x^4+256*g^2*x^3)*g'+12500*g^3*x^3-576*g^3*x^2)*g'''+(-(9375/4)*g^2*x^4+192*g^2*x^3)*g''^2+(((28125/4)*g*x^4-576*g*x^3)*(g')^2+(-18750*g^2*x^3+864*g^2*x^2)*g'+22500*g^3*x^2-408*g^3*x)*g''+(-(46875/16)*x^4+240*x^3)*(g')^4+(9375*g*x^3-432*g*x^2)*(g')^3+(-11250*g^2*x^2+204*g^2*x)*(g')^2+(7500*g^3*x-12*g^3)*g'+120*g^4 = 0. - Robert Israel, Jul 16 2015
a(n) = [x^n] 1/((1-5*x) * (1-x)^(4*n+1)). - Seiichi Manyama, Aug 03 2025
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} 5^k * (-4)^(n-k) * binomial(5*n+1,k) * binomial(5*n-k,n-k).
G.f.: g^2/(5-4*g)^2 where g = 1+x*g^5 is the g.f. of A002294. (End)

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

A079589 a(n) = C(5*n+1,n).

Original entry on oeis.org

1, 6, 55, 560, 5985, 65780, 736281, 8347680, 95548245, 1101716330, 12777711870, 148902215280, 1742058970275, 20448884000160, 240719591939480, 2840671544105280, 33594090947249085, 398039194165652550, 4724081931321677925, 56151322242892212960, 668324943343021950370

Offset: 0

Benoit Cloitre, Jan 26 2003

nonn

a(n) is the number of paths from (0,0) to (5n,n) taking north and east steps while avoiding exactly 2 consecutive north steps. - Shanzhen Gao, Apr 15 2010

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

Cf. A052203.
Cf. A001449, A079678, A371753, A385632, A386812.
Cf. A005810, A183160.

[Binomial(5*n+1, n): n in [0..20]]; // Vincenzo Librandi, Aug 07 2014

seq(binomial(5*n+1,n),n=0..100); # Robert Israel, Aug 07 2014

Table[Binomial[5n+1,n],{n,0,20}]  (* Harvey P. Dale, Jan 23 2011 *)

a(n) is asymptotic to c*(3125/256)^n/sqrt(n) with c=0.557.... [c = 5^(3/2)/(sqrt(Pi)*2^(7/2)) = 0.55753878629774... - Vaclav Kotesovec, Feb 14 2019 and Aug 20 2025]
8*n*(4*n+1)*(2*n-1)*(4*n-1)*a(n) -5*(5*n+1)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1)=0. - R. J. Mathar, Jul 17 2014
G.f.: hypergeom([2/5, 3/5, 4/5, 6/5], [1/2, 3/4, 5/4], (3125/256)*x). - Robert Israel, Aug 07 2014
a(n) = [x^n] 1/(1 - x)^(2*(2*n+1)). - Ilya Gutkovskiy, Oct 10 2017
From Seiichi Manyama, Aug 16 2025: (Start)
a(n) = Sum_{k=0..n} binomial(5*n-k,n-k).
G.f.: 1/(1 - x*g^3*(5+g)) where g = 1+x*g^5 is the g.f. of A002294.
G.f.: g^2/(5-4*g) where g = 1+x*g^5 is the g.f. of A002294.
G.f.: B(x)^2/(1 + 4*(B(x)-1)/5), where B(x) is the g.f. of A001449. (End)

A371753 a(n) = Sum_{k=0..floor(n/2)} binomial(5n-2k-1,n-2*k).

Original entry on oeis.org

1, 4, 37, 376, 4013, 44064, 492871, 5585080, 63901421, 736575316, 8540549322, 99503540008, 1163910870767, 13660217796736, 160782910480936, 1897131524755896, 22433316399634669, 265775992115557076, 3154067508987675679, 37487016824453703920, 446148092364247390618

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Cf. A026641, A147855, A183160.

Cf. A001449, A079589, A079678, A385632, A386812.

A371753 := proc(n)
    add( binomial(5*n-2*k-1,n-2*k),k=0..floor(n/2)) ;
end proc:
seq(A371753(n),n=0..50) ; # R. J. Mathar, Sep 27 2024

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

a(n) = [x^n] 1/((1-x^2) * (1-x)^(4*n)).
a(n) ~ 5^(5*n + 3/2) / (3 * sqrt(Pi*n) * 2^(8*n + 5/2)). - Vaclav Kotesovec, Apr 05 2024
Conjecture D-finite with recurrence +1024*n*(796184150374453*n -1374782084855770) *(4*n-3)*(2*n-1)*(4*n-1)*a(n) +64*(-4720591427354845074*n^5 +16046598674673412696*n^4 -14164434258362644374*n^3 -6132680339747354209*n^2 +16406971563067867560*n -7312237120275595200)*a(n-1) +40*(-4968388566264801507*n^5 +51044954667717039608*n^4 -218029351288077225930*n^3 +471970442274586326109*n^2 -511707487331990011785*n +221366817798624198360)*a(n-2) -25*(5*n-11) *(719005061479699*n -1438086256867727)*(5*n-9) *(5*n-13)*(5*n-12)*a(n-3)=0. - R. J. Mathar, Sep 27 2024
From Seiichi Manyama, Aug 05 2025: (Start)
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(5*n+1,k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(4*n+k,k). (End)
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(5*n+1,k) * binomial(5*n-k,n-k).
G.f.: g^2/((-1+2*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. (End)
G.f.: B(x)^2/(1 + 6*(B(x)-1)/5), where B(x) is the g.f. of A001449. - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^3*(-5+9*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 16 2025

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

Original entry on oeis.org

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)

A386897 a(n) = 4^n * binomial(5*n/2,n).

Original entry on oeis.org

1, 10, 160, 2860, 53760, 1040060, 20500480, 409404600, 8255569920, 167718033340, 3427543285760, 70384350760360, 1451115518361600, 30018413447053080, 622759359440486400, 12951795276279787760, 269947721071617638400, 5637113741080428839100

Offset: 0

Seiichi Manyama, Aug 07 2025

Paolo Xausa, Table of n, a(n) for n = 0..700

Cf. A386812, A386895, A386896, A386898.

Cf. A000302, A098430, A244038.

Table[Sum[2^k *(-1)^(n-k)*Binomial[5*n+1, k]*Binomial[2*n-k, n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *)
A386897[n_] := 4^n*Binomial[5*n/2, n]; Array[A386897, 20, 0] (* Paolo Xausa, Aug 26 2025 *)

PARI
```
a(n) = 4^n*binomial(5*n/2, n);
```

a(n) == 0 (mod 10) for n > 0.
a(n) = Sum_{k=0..n} binomial(5*n+1,k) * binomial(4*n-k,n-k).
a(n) = [x^n] (1+x)^(5*n+1)/(1-x)^(3*n+1).
a(n) = [x^n] 1/((1-x)^(n+1) * (1-2*x)^(3*n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n+k,k) * binomial(2*n-k,n-k).
a(n) = [x^n] 1/(1-4*x)^(3*n/2+1).
a(n) = [x^n] (1+4*x)^(5*n/2).
a(n) ~ 2^(n - 1/2) * 5^((5*n+1)/2) / (sqrt(Pi*n) * 3^((3*n+1)/2)). - Vaclav Kotesovec, Aug 07 2025
D-finite with recurrence 3*n*(n-1)*(3*n-4) *(3*n-2)*a(n) -20*(5*n-4) *(5*n-8)*(5*n-2) *(5*n-6)*a(n-2)=0. - R. J. Mathar, Aug 21 2025
O.g.f.: hypergeom([1/5, 2/5, 3/5, 4/5], [1/3, 1/2, 2/3], (12500*x^2)/27) + 10*x*hypergeom([7/10, 9/10, 11/10, 13/10], [5/6, 7/6, 3/2], (12500*x^2)/27). - Karol A. Penson, Aug 26 2025

A386895 a(n) = Sum_{k=0..n} binomial(5n+1,k) binomial(2*n-k,n-k).

Original entry on oeis.org

1, 8, 94, 1220, 16590, 231808, 3297154, 47490696, 690461070, 10111370720, 148929775544, 2203898519732, 32741261744802, 488010179737920, 7294326822378060, 109294796958693520, 1641111255497600910, 24688289062391137056, 372020649062760239080, 5614219481885985162960

Offset: 0

Seiichi Manyama, Aug 07 2025

nonn

Cf. A386812, A386896, A386897, A386898.

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

a(n) = [x^n] (1+x)^(5*n+1)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x)^(3*n+1) * (1-2*x)^(n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k) * binomial(4*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(4*n-k,n-k).
a(n) = binomial(2*n, n)*hypergeom([-1-5*n, -n], [-2*n], -1). - Stefano Spezia, Aug 07 2025
D-finite with recurrence +135*n*(n-1)*(3*n-1)*(3*n-2)*a(n) +3*(n-1)*(104049*n^3 -434754*n^2 +745789*n -439424)*a(n-1) +36*(517211*n^4 -4353801*n^3 +13137926*n^2 -17477238*n +8846684)*a(n-2) +16*(-11442763*n^4 +46270475*n^3 +85309279*n^2 -584322689*n +652846590)*a(n-3) -4585920*(5*n-16) *(5*n-14) *(5*n-18)*(5*n-17)*a(n-4)=0. - R. J. Mathar, Aug 21 2025

A386896 a(n) = Sum_{k=0..n} binomial(5n+1,k) binomial(3*n-k,n-k).

Original entry on oeis.org

1, 9, 125, 1932, 31365, 523809, 8910356, 153544680, 2671398309, 46822319115, 825501663525, 14623742203200, 260088366645900, 4641248247561324, 83059406374007720, 1490097583932329232, 26790218420643034533, 482571492068274975135, 8707190579448431827991

Offset: 0

Seiichi Manyama, Aug 07 2025

nonn

Cf. A386812, A386895, A386897, A386898.

a(n) = sum(k=0, n, binomial(5*n+1, k)*binomial(3*n-k, n-k));

a(n) = [x^n] (1+x)^(5*n+1)/(1-x)^(2*n+1).
a(n) = [x^n] 1/((1-x) * (1-2*x))^(2*n+1).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k) * binomial(3*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(2*n+k,k) * binomial(3*n-k,n-k).
a(n) = binomial(3*n, n)*hypergeom([-1-5*n, -n], [-3*n], -1). - Stefano Spezia, Aug 07 2025
D-finite with recurrence 202*n*(n-1)*(2*n-1)*(2*n-3)*a(n) -3*(n-1)*(2*n-3) *(14093*n^2-15245*n+5226)*a(n-1) +4*(355081*n^4 -1597876*n^3 +2789549*n^2 -2405270*n+926160)*a(n-2) -3840*(5*n-11)*(5*n-9) *(5*n-13)*(5*n-12)*a(n-3)=0. - R. J. Mathar, Aug 21 2025

A386898 a(n) = Sum_{k=0..n} binomial(5n+1,k) binomial(5*n-k,n-k).

Original entry on oeis.org

1, 11, 199, 4031, 85919, 1885311, 42154111, 955020287, 21847988735, 503573013503, 11675986431999, 272033089535999, 6363380561141759, 149354395882487807, 3515589114309115903, 82957940541503045631, 1961823306198598418431, 46482660516543479939071

Offset: 0

Seiichi Manyama, Aug 07 2025

nonn

Cf. A386812, A386895, A386896, A386897.

Cf. A178792, A383326, A383716.

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

a(n) = [x^n] (1+x)^(5*n+1)/(1-x)^(4*n+1).
a(n) = [x^n] 1/((1-x) * (1-2*x)^(4*n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n+k,k).
a(n) = binomial(5*n, n)*hypergeom([-1-5*n, -n], [-5*n], -1). - Stefano Spezia, Aug 07 2025

cp's OEIS Frontend

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

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A079678 a(n) = a(n,m) = Sum_{k=0..n} binomial(m*k,k)*binomial(m*(n-k),n-k) for m=5.

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A160906 Row sums of A159841.

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A079589 a(n) = C(5*n+1,n).

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A371753 a(n) = Sum_{k=0..floor(n/2)} binomial(5*n-2*k-1,n-2*k).

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

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A386897 a(n) = 4^n * binomial(5*n/2,n).

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A386895 a(n) = Sum_{k=0..n} binomial(5*n+1,k) * binomial(2*n-k,n-k).

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A079678 a(n) = a(n,m) = Sum_{k=0..n} binomial(mk,k)binomial(m*(n-k),n-k) for m=5.

A371753 a(n) = Sum_{k=0..floor(n/2)} binomial(5n-2k-1,n-2*k).

A386895 a(n) = Sum_{k=0..n} binomial(5n+1,k) binomial(2*n-k,n-k).

A386896 a(n) = Sum_{k=0..n} binomial(5n+1,k) binomial(3*n-k,n-k).

A386898 a(n) = Sum_{k=0..n} binomial(5n+1,k) binomial(5*n-k,n-k).