A160906 -id:A160906 - 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

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

Original entry on oeis.org

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

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

Original entry on oeis.org

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

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

A244038 a(n) = 4^n * binomial(3*n/2,n).

Original entry on oeis.org

1, 6, 48, 420, 3840, 36036, 344064, 3325608, 32440320, 318704100, 3148873728, 31256180280, 311452237824, 3113596420200, 31213674823680, 313672599360720, 3158823892156416, 31870058661517860, 322076161553203200, 3259691964853493400, 33034843349204336640, 335189468043077792760

Offset: 0

N. J. A. Sloane, Jun 28 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
I. M. Gessel, A short proof of the Deutsch-Sagan congruence for connected non crossing graphs, arXiv preprint arXiv:1403.7656 [math.CO], 2014. See f_1(n).

Cf. A045741, A244039.
Cf. A160906, A383326.
Cf. A000302, A098430, A386897.

[Round(4^n*Gamma(3*n/2+1)/(Gamma(n+1)*Gamma(n/2+1))): n in [0..40]]; // G. C. Greubel, Aug 06 2018

f1:=n->4^n*binomial(3*n/2,n); [seq(f1(n),n=0..40)];

Table[4^n Binomial[3 n/2, n], {n, 0, 40}] (* Vincenzo Librandi, Jun 29 2014 *)

{a(n) = if( n<1, n==0, 3^n * polcoeff( serreverse( x / (x+1) / 2 * sqrt((x+3) / (x+1) / 3 + x * O(x^n))), n))}; /* Michael Somos, Jan 27 2018 */

a(n) = A045741(n+1) + A244039(n) [Gessel].
a(n) = [x^n] 1/sqrt(1 - 4*x)^(n+2). - Ilya Gutkovskiy, Oct 10 2017
G.f. A(x) satisfies: A(x)^3 * (1 - 108*x^2) = 3*A(x) - 2. - Michael Somos, Jan 27 2018
a(n) = [x^n] ( (1 + 4*x)^(3/2) )^n. It follows that the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. - Peter Bala, Mar 05 2022
G.f.: 2/(1-2*sin(arcsin(216*x^2-1)/3)). - Vladimir Kruchinin, Oct 06 2022
G.f.: ((3^(5/6)*i + 3^(1/3))*(-18*i*z + sqrt(-324*z^2 + 3))^(1/3) - (3^(5/6)*i - 3^(1/3))*(18*i*z + sqrt(-324*z^2 + 3))^(1/3))/(2*sqrt(-324*z^2 + 3)), where i = sqrt(-1) is the imaginary unit. - Karol A. Penson, Oct 24 2024
From Seiichi Manyama, Aug 07 2025: (Start)
a(n) = Sum_{k=0..n} binomial(3*n+1,k) * binomial(2*n-k,n-k).
a(n) = [x^n] (1+x)^(3*n+1)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x) * (1-2*x))^(n+1).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(2*n-k,n-k). (End)

A226751 G.f.: 1 / (1 + 6xG(x) - 7xG(x)^2), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 9, 48, 289, 1761, 10932, 68664, 435201, 2777763, 17829489, 114968052, 744178716, 4832624044, 31469746632, 205422018288, 1343734578561, 8806130111847, 57805893969531, 380013533789928, 2501507255441049, 16486378106441697, 108773240389894056

Offset: 0

Paul D. Hanna, Jun 16 2013

nonn

			G.f.: A(x) = 1 + x + 9*x^2 + 48*x^3 + 289*x^4 + 1761*x^5 + 10932*x^6 +...
A related series is G(x) = 1 + x*G(x)^3, where
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
G(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
such that A(x) = 1/(1 + 6*x*G(x) - 7*x*G(x)^2).

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

Cf. A147855, A001764.
Cf. A005809, A006256, A160906, A183160.
Cf. A226705, A226733.

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

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

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

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

a(n) = Sum_{k=0..n} C(2*k, n-k) * C(3*n-2*k, k).
a(n) = Sum_{k=0..n} C(n+2*k, n-k) * C(2*n-2*k, k).
a(n) = Sum_{k=0..n} C(2*n+2*k, n-k) * C(n-2*k, k).
a(n) = Sum_{k=0..n} C(3*n+2*k, n-k) * C(-2*k, k).
G.f.: 1/(1 - x*G(x) - 7*x^2*G(x)^4), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
a(n) ~ 3^(3*n+3/2)/(5*sqrt(Pi*n)*2^(2*n+1)). - Vaclav Kotesovec, Jun 17 2013
Conjecture: 18*n*(2*n-1)*(55*n-76)*a(n) +(-11605*n^3+28521*n^2-20870*n+4536)*a(n-1) -24*(55*n-21)*(3*n-4)*(3*n-2)*a(n-2)=0. - R. J. Mathar, Jun 14 2016
From Seiichi Manyama, Aug 05 2025: (Start)
a(n) = [x^n] 1/((1+2*x) * (1-x)^(2*n+1)).
a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(3*n+1,k).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(2*n+k,k). (End)
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k).
G.f.: G(x)^2/((-2+3*G(x)) * (3-2*G(x))) where G(x) = 1+x*G(x)^3 is the g.f. of A001764. (End)
G.f.: B(x)^2/(1 + 5*(B(x)-1)/3), where B(x) is the g.f. of A005809. - Seiichi Manyama, Aug 15 2025

A387007 a(n) = Sum_{k=0..n} binomial(3*n+2,k).

Original entry on oeis.org

1, 6, 37, 232, 1471, 9402, 60460, 390656, 2533987, 16489546, 107594213, 703680424, 4611412196, 30273024984, 199045392232, 1310535994368, 8639411571051, 57017083602138, 376674527189599, 2490742704227192, 16483857933928471, 109175823528400778, 723611538997758784

Offset: 0

Seiichi Manyama, Aug 12 2025

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

Cf. A066380, A160906, A387008.

Cf. A001764.

[&+[Binomial(3*n+2,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025

Table[Sum[Binomial[3*n+2,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 27 2025 *)

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

a(n) = [x^n] (1+x)^(3*n+2)/(1-x).
a(n) = [x^n] 1/((1-x)^(2*n+2) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+2,k) * binomial(3*n-k+1,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k+1,n-k).
G.f.: g^3/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
D-finite with recurrence: 24*(5*n+1)*(3*n-1)*(3*n-2)*a(n-2) -(295*n^3-156*n^2-61*n+6)*a(n-1) +2*n*(2*n+1)*(5*n-4)*a(n). - Georg Fischer, Aug 17 2025

A387008 a(n) = Sum_{k=0..n} binomial(3*n+3,k).

Original entry on oeis.org

1, 7, 46, 299, 1941, 12616, 82160, 536155, 3505699, 22964087, 150676186, 990134948, 6515349244, 42925973608, 283134975936, 1869455684187, 12355133446527, 81725384344741, 541021064605298, 3584203906519219, 23761237400402597, 157623924396214756, 1046244086051121248

Offset: 0

Seiichi Manyama, Aug 12 2025

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

Cf. A066380, A160906, A387007.

Cf. A001764.

[&+[Binomial(3*n+3,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025

Table[Sum[Binomial[3*n+3,k], {k,0,n}], {n,0,25}] (* Vaclav Kotesovec, Aug 20 2025 *)

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

a(n) = [x^n] (1+x)^(3*n+3)/(1-x).
a(n) = [x^n] 1/((1-x)^(2*n+3) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+3,k) * binomial(3*n-k+2,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k+2,n-k).
G.f.: g^4/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
D-finite with recurrence: 24*(3*n-2)*(3*n-1)*(5*n^2+n-2)*a(n-2) -(295*n^4-156*n^3-339*n^2+12*n+20)*a(n-1) +2*(2*n+1)*(n+1)*(5*n^2-9*n+2)*a(n) = 0. - Georg Fischer, Aug 17 2025
a(n) ~ 3^(3*n + 7/2) / (sqrt(Pi*n) * 2^(2*n+3)). - Vaclav Kotesovec, Aug 20 2025

A387033 a(n) = Sum_{k=0..n} binomial(3*n-1,k).

Original entry on oeis.org

1, 3, 16, 93, 562, 3473, 21778, 137980, 880970, 5658537, 36519556, 236618693, 1538132224, 10026362492, 65513177704, 428957009288, 2813768603466, 18486790962201, 121634649321208, 801330506737399, 5285305708097522, 34896814868837161, 230631268849574378

Offset: 0

Seiichi Manyama, Aug 13 2025

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

Cf. A066380, A160906, A385823, A386006, A387007, A387008.
Cf. A114121, A387037.
Cf. A001764, A047099, A165817.

[&+[Binomial(3*n-1,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025

Table[Sum[Binomial[3*n-1,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 27 2025 *)

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

a(n) = [x^n] (1+x)^(3*n-1)/(1-x).
a(n) = [x^n] 1/((1-x)^(2*n-1) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n-1,k) * binomial(3*n-k-2,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k-2,n-k).
G.f.: 1/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
D-finite with recurrence: 24*(5*n+6)*(3*n-4)*(3*n-5)*a(n-2)-(295*n^3-451*n^2-234*n+360)*a(n-1)+2*n*(5*n+1)*(2*n-3)*a(n) = 0. - Georg Fischer, Aug 17 2025
a(n) ~ 3^(3*n - 1/2) / (sqrt(Pi*n) * 2^(2*n-1)). - Vaclav Kotesovec, Aug 27 2025

A385823 a(n) = Sum_{k=0..n} binomial(3*n-3,k).

Original entry on oeis.org

1, 1, 7, 42, 256, 1586, 9949, 63004, 401930, 2579130, 16628809, 107636402, 699030226, 4552602248, 29722279084, 194458630304, 1274628824490, 8368726082346, 55027110808177, 362301656545966, 2388274575638228, 15760514137668514, 104108685843640517, 688331413734386356

Offset: 0

Seiichi Manyama, Aug 13 2025

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

Cf. A066380, A160906, A386006, A387007, A387008, A387033.

Cf. A001764, A047099, A165817.

[&+[Binomial(3*n-3,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025

Table[Sum[Binomial[3*n-3,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 27 2025 *)

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

a(n) = [x^n] (1+x)^(3*n-3)/(1-x).
a(n) = [x^n] 1/((1-x)^(2*n-3) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n-3,k) * binomial(3*n-k-4,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k-4,n-k).
G.f.: 1/(g^2 * (2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.

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

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

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

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A244038 a(n) = 4^n * binomial(3*n/2,n).

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A226751 G.f.: 1 / (1 + 6*x*G(x) - 7*x*G(x)^2), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

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

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

A226751 G.f.: 1 / (1 + 6xG(x) - 7xG(x)^2), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.