A141223 -id:A141223 - 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

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

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)

A386957 a(n) = Sum_{k=0..n} 8^k * binomial(2*n+1,n-k).

Original entry on oeis.org

1, 11, 114, 1163, 11806, 119646, 1211820, 12271179, 124251318, 1258065866, 12737997724, 128972535582, 1305848105836, 13221716621852, 133869898347264, 1355432788629963, 13723757247851046, 138953043155444562, 1406899565919247884, 14244858120395937738, 144229188529316725956

Offset: 0

Seiichi Manyama, Aug 11 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A000302, A000984, A026641, A141223, A377011.

Cf. A386958, A386960.

[&+[8^k * Binomial(2*n+1, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 14 2025

Table[Sum[8^k*Binomial[2*n+1,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 14 2025 *)

a(n) = sum(k=0, n, 8^k*binomial(2*n+1, n-k));

a(n) = [x^n] 1/((1-9*x) * (1-x)^(n+1)).
a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 9^k * binomial(2*n-k,n-k).
G.f.: 2/( sqrt(1-4*x) * (9*sqrt(1-4*x)-7) ).
D-finite with recurrence 8*n*a(n) +(-113*n+16)*a(n-1) +162*(2*n-1)*a(n-2)=0. - R. J. Mathar, Aug 21 2025

A377011 a(n) = Sum_{k=0..n} 3^k * binomial(2*n+1,n-k).

Original entry on oeis.org

1, 6, 34, 188, 1026, 5556, 29940, 160824, 862018, 4613636, 24667644, 131795912, 703812916, 3757135752, 20051429544, 106992663408, 570827898306, 3045193326372, 16244056119084, 86646747723048, 462161936699196, 2465043081687192, 13147597801986264, 70123266087502608

Offset: 0

Seiichi Manyama, Aug 11 2025

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

Cf. A293490, A383832, A386942.

Cf. A000302, A000984, A026641, A141223, A386957.

[&+[3^k * Binomial(2*n+1,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025

Table[Sum[3^k * Binomial[2*n+1,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *)

a(n) = sum(k=0, n, 3^k*binomial(2*n+1, n-k));

a(n) = [x^n] 1/((1-4*x) * (1-x)^(n+1)).
a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 4^k * binomial(2*n-k,n-k).
G.f.: 1/( sqrt(1-4*x) * (2*sqrt(1-4*x)-1) ).
a(n) ~ 2^(4*n+2) / 3^(n+1). - Vaclav Kotesovec, Aug 20 2025
D-finite with recurrence 3*n*a(n) +2*(-14*n+3)*a(n-1) +32*(2*n-1)*a(n-2)=0. - R. J. Mathar, Aug 21 2025

A387085 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(2*n+1,k).

Original entry on oeis.org

1, 0, 4, 8, 36, 120, 456, 1680, 6340, 23960, 91224, 348656, 1337896, 5149872, 19877904, 76907808, 298176516, 1158168792, 4505865144, 17555689008, 68490100536, 267518448912, 1046041377264, 4094231982048, 16039426479336, 62887835652720, 246761907761776, 968943740083040

Offset: 0

Seiichi Manyama, Aug 16 2025

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

Cf. A000302, A000984, A001700, A026641, A141223, A377011, A386957.
Cf. A226733, A226705, A226751.
Cf. A005810, A079589, A183160.

[&+[(-3)^(n-k) * Binomial(2*n+1,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025

Table[Sum[(-3)^(n-k)*Binomial[2*n+1,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *)

a(n) = sum(k=0, n, (-3)^(n-k)*binomial(2*n+1, k));

a(n) = [x^n] (1+x)^(2*n+1)/(1+3*x).
a(n) = [x^n] 1/((1-x)^(n+1) * (1+2*x)).
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} (-2)^k * binomial(2*n-k,n-k).
G.f.: 1/( 4*x - 1 + 2*sqrt(1 - 4*x) ).
G.f.: 1/(1 - 4*x*(-1+g)) where g = 1+x*g^2 is the g.f. of A000108.
G.f.: g^2/((-2+3*g) * (2-g)) where g = 1+x*g^2 is the g.f. of A000108.
G.f.: B(x)^2/(1 + 2*(B(x)-1)), where B(x) is the g.f. of A000984.
D-finite with recurrence 3*n*a(n) +2*(-4*n+3)*a(n-1) +8*(-2*n+1)*a(n-2)=0. - R. J. Mathar, Aug 19 2025

A188481 Riordan matrix (1/(1-4x),(1-sqrt(1-4x))/(2*sqrt(1-4x))).

Original entry on oeis.org

1, 4, 1, 16, 7, 1, 64, 38, 10, 1, 256, 187, 69, 13, 1, 1024, 874, 406, 109, 16, 1, 4096, 3958, 2186, 748, 158, 19, 1, 16384, 17548, 11124, 4570, 1240, 216, 22, 1, 65536, 76627, 54445, 25879, 8485, 1909, 283, 25, 1, 262144, 330818, 259006, 138917, 52984, 14471, 2782, 359, 28, 1

Offset: 0

Emanuele Munarini, Apr 01 2011

Row sums = A141223;
Diagonal sums = A188482;
Inverse matrix: (1/(1+2x)^2, x(1+x)/(1+2x)^2).

			Triangle begins:
      1;
      4,     1;
     16,     7,     1;
     64,    38,    10,     1;
    256,   187,    69,    13,    1;
   1024,   874,   406,   109,   16,    1;
   4096,  3958,  2186,   748,  158,   19,   1;
  16384, 17548, 11124,  4570, 1240,  216,  22,  1;
  65536, 76627, 54445, 25879, 8485, 1909, 283, 25, 1;

Cf. A141223, A188482.

Flatten[Table[Sum[Binomial[n+i,n]Binomial[n-i,k]2^(n-k-i),{i,0,n-k}],{n,0,8},{k,0,8}]]

create_list(sum(binomial(n+i,n)*binomial(n-i,k)*2^(n-k-i),i,0,n-k),n,0,8,k,0,n);

T(n,k) = [x^n] ((1-sqrt(1-4*x))/(2*sqrt(1-4*x)))^k/(1-4*x).

Recurrence: T(n+1,k+1) = T(n,k) + 3*T(n,k-1) + T(n,k-2) - T(n,k-3) + T(n,k-4) - T(n,k-5) + ...

Comment corrected by Philippe Deléham, Jan 22 2014

A188482 Diagonal sums of the Riordan matrix (1/(1-4x),(1-sqrt(1-4x))/(2*sqrt(1-4x))) (A188481).

Original entry on oeis.org

1, 4, 17, 71, 295, 1221, 5040, 20761, 85380, 350659, 1438568, 5896098, 24145941, 98812861, 404118745, 1651811920, 6748282361, 27556753703, 112482005583, 458958881572, 1872034052651, 7633342954234, 31116252892098, 126806214027741, 516633711969649

Offset: 0

Emanuele Munarini, Apr 01 2011

Cf. A141223, A188481.

Table[Sum[Binomial[k,i]Binomial[2n+k-2i+2,n-k](-1)^i,{k,0,n},{i,0,k}],{n,0,12}]

makelist(sum(sum(binomial(k,i)*binomial(2*n+k-2*i+2,n-k)*(-1)^i,i,0,k),k,0,n),n,0,12);

a(n) = [x^n] 1/((1-x)^(n+1)*(1-2*x-x^2+x^3)).
a(n) = Sum_{k=0..n} Sum_{i=0..k} binomial(k,i)*binomial(2*n+k-2*i+2, n-k)*(-1)^i.
G.f.: (2 - 7*x - 4*x^2 + x*sqrt(1-4*x))/(2 - 14*x + 16*x^2 + 30*x^3 + 8*x^4).
Conjecture: (-n+1)*a(n) + (7*n-9)*a(n-1) + 2*(-4*n+7)*a(n-2) + (-15*n+23)*a(n-3) + 2*(-2*n+3)*a(n-4) = 0. - R. J. Mathar, Jun 14 2016

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

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

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A386957 a(n) = Sum_{k=0..n} 8^k * binomial(2*n+1,n-k).

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A377011 a(n) = Sum_{k=0..n} 3^k * binomial(2*n+1,n-k).

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A387085 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(2*n+1,k).

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A188481 Riordan matrix (1/(1-4x),(1-sqrt(1-4x))/(2*sqrt(1-4x))).

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A188482 Diagonal sums of the Riordan matrix (1/(1-4x),(1-sqrt(1-4x))/(2*sqrt(1-4x))) (A188481).

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