A079678 -id:A079678 - cp's OEIS frontend

A006256 a(n) = Sum_{k=0..n} binomial(3k,k)binomial(3n-3k,n-k).

1, 6, 39, 258, 1719, 11496, 77052, 517194, 3475071, 23366598, 157206519, 1058119992, 7124428836, 47983020624, 323240752272, 2177956129818, 14677216121871, 98923498131762, 666819212874501, 4495342330033938, 30308036621747679, 204356509814519712

Offset: 0

N. J. A. Sloane, Don Knuth

The right-hand sides of several of the "Ruehr identities". - N. J. A. Sloane, Feb 20 2020

Convolution of A005809 with itself. - Emeric Deutsch, May 22 2003

Allouche, J-P. "Two binomial identities of Ruehr Revisited." The American Mathematical Monthly 126.3 (2019): 217-225.
Alzer, Horst, and Helmut Prodinger. "On Ruehr's Identities." Ars Comb. 139 (2018): 247-254.
Bai, Mei, and Wenchang Chu. "Seven equivalent binomial sums." Discrete Mathematics 343.2 (2020): 111691.
M. Petkovsek et al., A=B, Peters, 1996, p. 165.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Jean-Paul Allouche, A generalization of an identity due to Kimura and Ruehr, arXiv preprint arXiv:1706.08929 [math.NT], 2017.
Jean-Paul Allouche, Two exercises of Comtet and two identities of Ruehr, arXiv preprint arXiv:1707.05751 [math.NT], 2017.
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.
Shalosh B. Ekhad, Doron Zeilberger, Some Remarks on a recent article by J.-P. Allouche, arXiv:1903.09511 [math.CO], 2019.
N, Kimura and O. G. Ruehr, Change of variable formula for definite integral. Problem E2765, Am. Math. Mnthly, 87, 1980, 307-308.
S. Meehan, A. Tefera, M. Weselcouch, A. Zeleke, Proofs of Ruehr's identities, Integers 14 (2014) A10.
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Y_n for s=3).

Cf. A036829.
Cf. A005809, A160906, A183160, A226751.
Cf. A000302, A078995, A079563, A079678, A079679.
Cf. A141223, A385605, A385632.

a006256 n = a006256_list !! n
a006256_list = f (tail a005809_list) [1] where
   f (x:xs) zs = (sum $ zipWith (*) zs a005809_list) : f xs (x : zs)
-- Reinhard Zumkeller, Sep 21 2014

[&+[Binomial(3*k, k) *Binomial(3*n-3*k, n-k): k in [0..n]]:n in  [0..22]]; // Vincenzo Librandi, Feb 21 2020

a:= proc(n) option remember; `if`(n<2, 5*n+1,
      ((216*n^2-270*n+96) *a(n-1)
      -81*(3*n-2)*(3*n-4) *a(n-2)) /(n*(16*n-8)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 07 2012

a[n_] := HypergeometricPFQ[{1/3, 2/3, 1/2-n, -n}, {1/2, 1/3-n, 2/3-n}, 1]*(3n)!/(n!*(2n)!); Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 20 2012 *)
Table[Sum[Binomial[3k,k]Binomial[3n-3k,n-k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Oct 23 2013 *)

a(n)=sum(k=0,n, binomial(3*k,k)*binomial(3*n-3*k,n-k)) \\ Charles R Greathouse IV, Feb 07 2017

a = lambda n: binomial(3*n+1,n)*hypergeometric([1,-n],[2*n+2],-2)
[simplify(a(n)) for n in range(20)] # Peter Luschny, May 19 2015

a(n) = (3/4)*(27/4)^n*(1+c/sqrt(n)+o(n^(-1/2))) where c = (2/3)*sqrt(1/(3*Pi)) = 0.217156671956853298... More generally, a(n, m) = sum(k=0, n, C(m*k,k) *C(m*(n-k),n-k)) is asymptotic to (1/2)*m/(m-1)*(m^m/(m-1)^(m-1))^n. See A000302, A078995 for cases m=2 and 4. - Benoit Cloitre, Jan 26 2003, extended by Vaclav Kotesovec, Nov 06 2012
G.f.: 1/(1-3*z*g^2)^2, where g=g(z) is given by g=1+z*g^3, g(0)=1, i.e. (in Maple command) g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z). - Emeric Deutsch, May 22 2003
D-finite with recurrence: 6*(36*n^2-45*n+16)*a(n-1) - 81*(3*n-4)*(3*n-2)*a(n-2) - 8*n*(2*n-1)*a(n) = 0. - Vaclav Kotesovec, Oct 05 2012
From Rui Duarte and António Guedes de Oliveira, Feb 17 2013: (Start)
a(n) = sum(k=0, n, C(3*k+l,k)*C(3*(n-k)-l,n-k)) for every real number l.
a(n) = sum(k=0, n, 2^(n-k)*C(3n+1,k)).
a(n) = sum(k=0, n, 3^(n-k)*C(2n+k,k)).  (End)
From Akalu Tefera, Sean Meehan, Michael Weselcouch, and Aklilu Zeleke, May 11 2013: (Start)
a(n) = sum(k=0, 2n, (-3)^k*C(3n - k, n)).
a(n) = sum(k=0, 2n, (-4)^k*C(3n + 1, 2n - k)).
a(n) = sum(k=0, n, 3^k*C(3n - k, 2n)).
a(n) = sum(k=0, n, 2^k*C(3n + 1, n - k)).  (End)
a(n) = C(3*n+1,n)*Hyper2F1(1,-n,2*n+2,-2). - Peter Luschny, May 19 2015
a(n) = [x^n] 1/((1-3*x) * (1-x)^(2*n+1)). - Seiichi Manyama, Aug 03 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k). - Seiichi Manyama, Aug 07 2025

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

Original entry on oeis.org

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

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

A079679 a(n) = a(n,m) = Sum_{k=0..n} binomial(mk,k)binomial(m*(n-k),n-k) for m=6.

Original entry on oeis.org

1, 12, 168, 2424, 35400, 520236, 7674144, 113482584, 1681028136, 24932533800, 370144424376, 5499182587416, 81748907485248, 1215834858032820, 18090048027643200, 269246037610828656, 4008495234662771688, 59692297399976544120, 889090275714779739120, 13245013739104555683600

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.

Seiichi Manyama, Table of n, a(n) for n = 0..851
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. A000302, A006256, A078995, A079563, A079678.

Cf. A002295, A226705.

a(n) = sum(k=0,n,5^(n-k)*binomial(6*n+1,k));
vector(30, n, a(n-1)) \\  Altug Alkan, Sep 30 2015

a(n) = 3/5*(46656/3125)^n*(1+c/sqrt(n)+o(n^-1/2)) where c=0.388...
c = 8/(3*sqrt(15*Pi)) = 0.388461664210517... - Vaclav Kotesovec, May 25 2020
a(n) = Sum_{k=0..n} binomial(6*k+l,k)*binomial(6*(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} 5^(n-k) * binomial(6*n+1,k).
a(n) = Sum_{k=0..n} 6^(n-k) * binomial(5*n+k,k). (End)
G.f.: hypergeom([1/6, 1/3, 1/2, 2/3, 5/6],[1/5, 2/5, 3/5, 4/5],46656*x/3125)^2.  - Mark van Hoeij, Apr 19 2013
a(n) = [x^n] 1/((1-6*x) * (1-x)^(5*n+1)). - Seiichi Manyama, Aug 03 2025
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} 6^k * (-5)^(n-k) * binomial(6*n+1,k) * binomial(6*n-k,n-k).
G.f.: g^2/(6-5*g)^2 where g = 1+x*g^6 is the g.f. of A002295. (End)

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)

A386614 a(n) = Sum_{k=0..n-1} binomial(5k+1,k) binomial(5n-5k,n-k-1).

Original entry on oeis.org

0, 1, 16, 220, 2880, 36850, 465536, 5834852, 72744640, 903525715, 11191199200, 138323478980, 1706860996096, 21034268215120, 258934785258240, 3184696786012500, 39140208951032960, 480734044749851305, 5901368553964031600, 72410017973538837880, 888114187330722044800, 10888921795007470528060

Offset: 0

Seiichi Manyama, Jul 27 2025

nonn

Cf. A079678, A386367, A386566, A386613.

Cf. A006419, A386612, A386616, A386617.

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

G.f.: g^3 * (g-1)/(5-4*g)^2 where g=1+x*g^5.
G.f.: g/((1-g)^2 * (1-5*g)^2) where g*(1-g)^4 = x.
a(n) = Sum_{k=0..n-1} binomial(5*k+1+l,k) * binomial(5*n-5*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(5*n+2,k).
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(4*n+k+2,k).
D-finite with recurrence +35651584*n*(4*n+1)*(2*n+1)*(4*n-1)*a(n) +8192*(56348704*n^4-268019168*n^3+418502324*n^2-264019618*n+57303885)*a(n-1) +160*(-65524820000*n^4+314102050000*n^3-463341186250*n^2+159732814775*n+76118151939)*a(n-2) +62500*(660806875*n^4-1813661250*n^3-5080986250*n^2+20705993100*n-17279228304)*a(n-3) +308935546875*(5*n-11)*(5*n-14)*(5*n-13)*(5*n-17)*a(n-4)=0. - R. J. Mathar, Aug 10 2025

A079563 a(n) = a(n,m) = Sum_{k=0..n} binomial(mk,k)binomial(m*(n-k),n-k) for m=7.

Original entry on oeis.org

1, 14, 231, 3934, 67851, 1177974, 20531770, 358788696, 6281076123, 110103674128, 1931983053056, 33926800240578, 596145343139514, 10480467311987778, 184327560283768776, 3243034966775972144, 57074433199551436347

Offset: 0

Benoit Cloitre, Jan 26 2003

nonn

More generally, for m>=2, 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 * (1 + (2*m-4)/(3*sqrt(Pi*n*m*(m-1)/2))), extended by Vaclav Kotesovec, May 25 2020

See A000302, A006256, A078995 for cases m=2,3 and 4.

Seiichi Manyama, Table of n, a(n) for n = 0..802
Rui Duarte and António Guedes de Oliveira, Short note on the convolution of binomial coefficients, arXiv:1302.2100 [math.CO], 2013.
D. Merlini, R. Sprugnoli, and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344.

Cf. A006256, A078995, A079678, A079679.

Cf. A002296.

a(n) = (7/12)*(823543/46656)^n*(1+c/sqrt(n)+o(n^-1/2)) where c=0.41...
c = 10/(3*sqrt(21*Pi)) = 0.410387535383... - Vaclav Kotesovec, May 25 2020
From Rui Duarte and António Guedes de Oliveira, Feb 17 2013: (Start)
a(n) = Sum_{k=0..n} binomial(7*k+x,k)*binomial(7*(n-k)-x,n-k) for any real x.
a(n) = Sum_{k=0..n} 6^(n-k)*binomial(7*n+1,k).
a(n) = Sum_{k=0..n} 7^(n-k)*binomial(6*n+k,k). (End)
a(n) = [x^n] 1/((1-7*x) * (1-x)^(6*n+1)). - Seiichi Manyama, Aug 03 2025
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} 7^k * (-6)^(n-k) * binomial(7*n+1,k) * binomial(7*n-k,n-k).
G.f.: g^2/(7-6*g)^2 where g = 1+x*g^7 is the g.f. of A002296. (End)

A386367 a(n) = Sum_{k=0..n-1} binomial(5k,k) binomial(5n-5k-2,n-k-1).

Original entry on oeis.org

0, 1, 13, 163, 2021, 24930, 306655, 3765448, 46182101, 565939603, 6931070490, 84845250370, 1038235255415, 12700966517968, 155336699256808, 1899439862390640, 23222289820948405, 283872591297526505, 3469680960837171415, 42404345427419774621, 518193229118757697930

Offset: 0

Seiichi Manyama, Jul 19 2025

nonn

			(1/5) * log( Sum_{k>=0} binomial(5*k,k)*x^k ) = x + 13*x^2/2 + 163*x^3/3 + 2021*x^4/4 + 4986*x^5 + ...

Cf. A000302, A036829, A308523, A386368.
Cf. A079678, A386566, A386613, A386614.
Cf. A002294, A118971.

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

my(N=30, x='x+O('x^N), g=x*sum(k=0, N, binomial(5*k+3, k)/(k+1)*x^k)); concat(0, Vec(g*(1-g)/(1-5*g)^2))

G.f.: g*(1-g)/(1-5*g)^2 where g*(1-g)^4 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/5) * log( Sum_{k>=0} binomial(5*k,k)*x^k ).
G.f.: (g-1)/(5-4*g)^2 where g=1+x*g^5.
a(n) = Sum_{k=0..n-1} binomial(5*k-2+l,k) * binomial(5*n-5*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(5*n-1,k).
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(4*n+k-1,k).

cp's OEIS Frontend

A006256 a(n) = Sum_{k=0..n} binomial(3*k,k)*binomial(3*n-3*k,n-k).

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

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

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A079679 a(n) = a(n,m) = Sum_{k=0..n} binomial(m*k,k)*binomial(m*(n-k),n-k) for m=6.

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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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A386614 a(n) = Sum_{k=0..n-1} binomial(5*k+1,k) * binomial(5*n-5*k,n-k-1).

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A006256 a(n) = Sum_{k=0..n} binomial(3k,k)binomial(3n-3k,n-k).

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

A079679 a(n) = a(n,m) = Sum_{k=0..n} binomial(mk,k)binomial(m*(n-k),n-k) for m=6.

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

A386614 a(n) = Sum_{k=0..n-1} binomial(5k+1,k) binomial(5n-5k,n-k-1).

A079563 a(n) = a(n,m) = Sum_{k=0..n} binomial(mk,k)binomial(m*(n-k),n-k) for m=7.

A386367 a(n) = Sum_{k=0..n-1} binomial(5k,k) binomial(5n-5k-2,n-k-1).