A371753 -id:A371753 - cp's OEIS frontend

A183160 a(n) = Sum_{k=0..n} C(n+k,n-k)C(2n-k,k).

1, 2, 11, 62, 367, 2232, 13820, 86662, 548591, 3498146, 22436251, 144583496, 935394436, 6071718512, 39523955552, 257913792342, 1686627623151, 11050540084902, 72522925038257, 476669316338542, 3137209052543927, 20672732229560032, 136374124374593072, 900541325129687272

Offset: 0

Paul D. Hanna, Dec 27 2010

nonn

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 62*x^3 + 367*x^4 + 2232*x^5 +...
A(x)^(1/2) = 1 + x + 5*x^2 + 26*x^3 + 145*x^4 + 841*x^5 + 5006*x^6 +...+ A183161(n)*x^n +...
Given triangle A085478(n,k) = C(n+k,n-k), which begins:
  1;
  1,  1;
  1,  3,  1;
  1,  6,  5,  1;
  1, 10, 15,  7, 1;
  1, 15, 35, 28, 9, 1; ...
ILLUSTRATE formula a(n) = Sum_{k=0..n} A085478(n,k)*A085478(n,n-k):
a(2) = 11 = 1*1 + 3*3 + 1*1;
a(3) = 62 = 1*1 + 6*5 + 5*6 + 1*1;
a(4) = 367 = 1*1 + 10*7 + 15*15 + 7*10 + 1*1;
a(5) = 2232 = 1*1 + 15*9 + 35*28 + 28*35 + 9*15 + 1*1; ...

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

Cf. A001764, A085478, A183161, A199033, A218622.
Cf. A026641, A147855, A371753.
Cf. A005809, A006256, A160906, A226751.
Cf. A171822.

[(&+[Binomial(n+k, 2*k)*Binomial(2*n-k, k): k in [0..n]]): n in [0..25]]; // G. C. Greubel, Feb 22 2021

Table[Sum[Binomial[n+k,n-k]Binomial[2n-k,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jul 19 2011 *)
Table[HypergeometricPFQ[{-n, -n, 1/2 -n, n+1}, {1/2, 1, -2*n}, 1], {n, 0, 25}] (* G. C. Greubel, Feb 22 2021 *)

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

{a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(1/(1-2*x*G^2-3*x^2*G^4), n)} \\ Paul D. Hanna, Nov 03 2012

{a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(1/(1+3*x*G-5*x*G^2), n)} \\ Paul D. Hanna, Jun 16 2013
for(n=0, 30, print1(a(n), ", "))

a = lambda n: binomial(3*n+1,n)*hypergeometric([1,-n],[2*n+2],2)
[simplify(a(n)) for n in range(26)] # Peter Luschny, May 19 2015

a(n) = Sum_{k=0..n} A085478(n,k)*A085478(n,n-k).
Self-convolution of A183161 (an integer sequence):
a(n) = Sum_{k=0..n} A183161(k)*A183161(n-k).
a(n) = Sum_{k=0..n} binomial(2*n+k,k) * cos((n+k)*Pi). - Arkadiusz Wesolowski, Apr 02 2012
Recurrence: 320*n*(2*n-1)*a(n) = 8*(346*n^2 + 79*n - 327)*a(n-1) + 6*(1688*n^2-6241*n+5981)*a(n-2) + 261*(3*n-7)*(3*n-5)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 3^(3*n+3/2)/(2^(2*n+3)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012
...
G.f.: A(x) = 1/(1 - 2*x*G(x)^2 - 3*x^2*G(x)^4), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Nov 03 2012
G.f.: A(x) = 1 + x*d/dx { log( G(x)^5/(1+x*G(x)^2) )/2 }, where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Nov 04 2012
G.f.: A(x) = 1/(1 + 3*x*G(x) - 5*x*G(x)^2), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - Paul D. Hanna, Jun 16 2013
a(n) = C(3*n+1,n)*Hyper2F1([1,-n],[2*n+2],2). - Peter Luschny, May 19 2015
a(n) = [x^n] 1/((1 - x^2)*(1 - x)^(2*n)). - Ilya Gutkovskiy, Oct 25 2017
From G. C. Greubel, Feb 22 2021: (Start)
a(n) = Sum_{k=0..n} A171822(n, k).
a(n) = Hypergeometric 4F3([-n, -n, 1/2 -n, n+1], [1/2, 1, -2*n], 1). (End)
a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-2*k-1,n-2*k). - Seiichi Manyama, Apr 05 2024
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n+1,k). - Seiichi Manyama, Aug 03 2025
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k).
G.f.: g^2/((-1+2*g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. (End)
G.f.: B(x)^2/(1 + 4*(B(x)-1)/3), where B(x) is the g.f. of A005809. - Seiichi Manyama, Aug 15 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)

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

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)

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)

A386371 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(5*n+1,k).

Original entry on oeis.org

1, 3, 31, 317, 3399, 37418, 419229, 4756104, 54463335, 628197809, 7287712566, 84942987198, 993941174829, 11668806723876, 137378189197112, 1621322803014672, 19175540677541991, 227217662222902443, 2696878158795639549, 32057403690640189635, 381573145993865438254

Offset: 0

Seiichi Manyama, Aug 17 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A001449, A079589, A079678, A371753, A385632, A386812.
Cf. A226705, A226733, A226751, A387085.
Cf. A002294.

[&+[(-3)^(n-k) * Binomial(5*n+1,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025

Table[Sum[(-3)^(n-k)*Binomial[5*n+1,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *)

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

a(n) = [x^n] (1+x)^(5*n+1)/(1+3*x).
a(n) = [x^n] 1/((1-x)^(4*n+1) * (1+2*x)).
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(5*n+1,k) * binomial(5*n-k,n-k).
a(n) = Sum_{k=0..n} (-2)^k * binomial(5*n-k,n-k).
G.f.: 1/(1 - x*g^3*(-10+13*g)) where g = 1+x*g^5 is the g.f. of A002294.
G.f.: g^2/((-2+3*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294.
G.f.: B(x)^2/(1 + 7*(B(x)-1)/5), where B(x) is the g.f. of A001449.
D-finite with recurrence 648*n*(135551509682187347695*n -244103380745409504343) *(4*n-1)*(2*n-1)*(4*n-3)*a(n) +(-33979500619583537984836075*n^5 +130803893690808003041848009*n^4 -168380151442376797602371231*n^3 +62069291513227826684567999*n^2 +49760069127090078338544954*n -39530305857276050670355320)*a(n-1) +40*(-108999332467309598098777*n^5 -28981701912184019189355*n^4 -1554974299825191814369159*n^3 +13581461461293413639358363*n^2 -28599284433109723900055776*n +18909354537435947334628944)*a(n-2) +211200*(5*n-11) *(5*n-9)*(28440609019752807*n +93502568692163852)*(5*n-13)*(5*n-12)*a(n-3)=0. - R. J. Mathar, Aug 26 2025

cp's OEIS Frontend

A183160 a(n) = Sum_{k=0..n} C(n+k,n-k)*C(2*n-k,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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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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A079589 a(n) = C(5*n+1,n).

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

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A386371 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(5*n+1,k).

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A183160 a(n) = Sum_{k=0..n} C(n+k,n-k)C(2n-k,k).

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

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.