A165817 -id:A165817 - cp's OEIS frontend

A386006 a(n) = Sum_{k=0..n} binomial(3*n-2,k).

1, 2, 11, 64, 386, 2380, 14893, 94184, 600370, 3850756, 24821333, 160645504, 1043243132, 6794414896, 44360053772, 290244832992, 1902631226010, 12493030680180, 82153313341429, 540953389469312, 3566279609565226, 23536562549993228, 155489358646406149

Offset: 0

Seiichi Manyama, Aug 13 2025

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

Cf. A066380, A160906, A385823, A387007, A387008, A387033.

Cf. A001764, A047099, A165817.

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

A323324 Coefficients T(n,k) of x^ny^(n-k)z^k in function A = A(x,y,z) such that A = 1 + xBC, B = 1 + yCA, and C = 1 + zAB, as a triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 27, 27, 1, 1, 64, 200, 64, 1, 1, 125, 875, 875, 125, 1, 1, 216, 2835, 6272, 2835, 216, 1, 1, 343, 7546, 30870, 30870, 7546, 343, 1, 1, 512, 17472, 118272, 217800, 118272, 17472, 512, 1, 1, 729, 36450, 378378, 1146717, 1146717, 378378, 36450, 729, 1, 1, 1000, 70125, 1056000, 4879875, 8016008, 4879875, 1056000, 70125, 1000, 1, 1, 1331, 126445, 2647359, 17649060, 44088044, 44088044, 17649060, 2647359, 126445, 1331, 1, 1, 1728, 216216, 6086080, 56119635, 201636864, 306330752, 201636864, 56119635, 6086080, 216216, 1728, 1

Offset: 0

Paul D. Hanna, Jan 11 2019

Row sums equal A165817(n), the number of compositions of n into 2*n parts, for n >= 0.

Central terms equal 2*A165817(n)^2, for n >= 1.

			This triangle begins:
1;
1, 1;
1, 8, 1;
1, 27, 27, 1;
1, 64, 200, 64, 1;
1, 125, 875, 875, 125, 1;
1, 216, 2835, 6272, 2835, 216, 1;
1, 343, 7546, 30870, 30870, 7546, 343, 1;
1, 512, 17472, 118272, 217800, 118272, 17472, 512, 1;
1, 729, 36450, 378378, 1146717, 1146717, 378378, 36450, 729, 1;
1, 1000, 70125, 1056000, 4879875, 8016008, 4879875, 1056000, 70125, 1000, 1;
1, 1331, 126445, 2647359, 17649060, 44088044, 44088044, 17649060, 2647359, 126445, 1331, 1;
1, 1728, 216216, 6086080, 56119635, 201636864, 306330752, 201636864, 56119635, 6086080, 216216, 1728, 1; ...
ROW SUMS are
[1, 2, 10, 56, 330, 2002, 12376, 77520, 490314, ..., binomial(3*n-1, n), ...].
CENTRAL TERMS are
[1, 8, 200, 6272, 217800, 8016008, 306330752, ..., 2*binomial(3*n-1, n)^2, ...].

Paul D. Hanna, Table of n, a(n) for n = 0..1325 terms of this triangle as read by rows 0..50
Thomas Einolf, Robert Muth, and Jeffrey Wilkinson, Injectively k-colored rooted forests, arXiv:2107.13417 [math.CO], 2021.

Cf. A323325, A165817 (row sums).

{T(n,k) = my(A=1,B=1,C=1); for(i=0,n,
A = 1 + x*B*C +x*O(x^n);
B = 1 + y*A*C +y*O(y^n);
C = 1 + z*A*B +z*O(z^n));
polcoeff(polcoeff(polcoeff(A,n,x),n-k,y),k,z)}
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))

Sum_{k=0..n} T(n,k) = binomial(3*n-1, n) for n >= 0.
Sum_{k=0..n} k * T(n,k) = n * binomial(3*n-1, n-1), for n >= 0.
T(2*n,n) = 2 * binomial(3*n-1, n)^2 for n >= 1, with a(0) = 1.
T(n,k) = T(n,n-k) for k = 0..n, for n >= 0.
T(n,1) = n^3 for n >= 0.
T(n,2) = n^3*(n^2-1)*(2*n-3)/24 for n >= 0.

A348478 Number of compositions of n into exactly n nonnegative parts such that each positive i-th part has the same parity as i.

Original entry on oeis.org

1, 1, 1, 4, 7, 23, 55, 164, 407, 1235, 3051, 9432, 23431, 72989, 182624, 571384, 1436855, 4511979, 11387467, 35866100, 90782837, 286622226, 727226578, 2300578392, 5848776767, 18533394763, 47197285045, 149769168304, 381956145802, 1213526310665, 3098742448230

Offset: 0

Alois P. Heinz, Oct 20 2021

nonn

			a(0) = 1: [].
a(1) = 1: [1].
a(2) = 1: [0,2].
a(3) = 4: [1,2,0], [0,2,1], [3,0,0], [0,0,3].
a(4) = 7: [1,2,1,0], [1,0,1,2], [3,0,1,0], [1,0,3,0], [0,2,0,2], [0,4,0,0], [0,0,0,4].

Alois P. Heinz, Table of n, a(n) for n = 0..2169

Cf. A001700, A062200, A088218, A122514, A165817, A305161, A324969, A348476.

b:= proc(n, t) option remember; `if`(t=0, 1-signum(n),
      add(`if`(j=0 or (t-j)::even, b(n-j, t-1), 0), j=0..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..33);

b[n_, t_] := b[n, t] = If[t == 0, 1 - Sign[n],
     Sum[If[j == 0 || EvenQ[t - j], b[n - j, t - 1], 0], {j, 0, n}]];
a[n_] :=  b[n, n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Apr 14 2022, after Alois P. Heinz *)

A365771 a(n) = binomial(2n+1, n)/(2n+1) * binomial(3*n-1, n) for n >= 0.

Original entry on oeis.org

1, 2, 20, 280, 4620, 84084, 1633632, 33256080, 701149020, 15191562100, 336424047960, 7584833081280, 173575987821600, 4022766574898400, 94247674040476800, 2228957491057276320, 53150802525726081660, 1276661433215969608500, 30863850087221160009000

Offset: 0

Paul D. Hanna, Oct 10 2023

Equals the central terms of triangle A365770.
Conjectures: given A033042 is the sums of distinct powers of 5, then
(1) a(5*A033042(n)) == 4 (mod 5) for n > 0,
(2) a(5*A033042(n) + 1) == 2 (mod 5) for n > 0,
(3) a(n) == 0 (mod 5) for n > 0 except when n or n-1 equals 5*A033042(k) for some k >= 0.

Paolo Xausa, Table of n, a(n) for n = 0..500

Cf. A007004, A000108, A165817, A033042, A319578, A356770.

A365771[n_] := Binomial[2*n + 1, n]/(2*n + 1)*Binomial[3*n - 1, n];
Array[A365771, 20, 0] (* Paolo Xausa, Oct 12 2024 *)

{a(n) = binomial(2*n+1, n)/(2*n+1) * binomial(3*n-1, n)}
for(n=0,30,print1(a(n),", "))

from math import comb
def A365771(n): return comb(m:=(n<<1)+1,n)*comb(m+n-2,n)//m if n else 1 # Chai Wah Wu, Oct 11 2023

a(n) = A365770(2*n,n) for n >= 0.
a(n) = A000108(n) * A165817(n) for n >= 0.
a(n) = 2*A319578(n) = (2/3) * A007004(n) for n >= 1. - Peter Bala, Aug 25 2025

A373865 Sum over all compositions of 2n into n parts of the least common multiple of the parts.

Original entry on oeis.org

1, 2, 8, 50, 160, 892, 3794, 19560, 80112, 371948, 1614598, 7180494, 30794746, 134269410, 574754496, 2471353090, 10542096528, 45057428356, 191653403306, 814082549052, 3444043955350, 14537736838038, 61174002263338, 256838845740468, 1075631257186986

Offset: 0

Alois P. Heinz, Jun 19 2024

nonn

			a(3) = 50 = 2 + 6*6 + 3*4: 222, 123, 132, 213, 231, 312, 321, 114, 141, 411.

Alois P. Heinz, Table of n, a(n) for n = 0..75

Cf. A005430 (the same for sum), A165817 (the same for product), A181851, A260878 (the same for gcd).

a(n) = A181851(2n,n).

A254747 a(n) = (1 + Sum_{j=0..n} (C(n,j)C(3j-1,j))) / 2.

Original entry on oeis.org

1, 2, 8, 47, 312, 2162, 15311, 109965, 797824, 5833298, 42910998, 317224800, 2354712927, 17538747124, 131017428431, 981194304302, 7364370502896, 55380344444150, 417176211054422, 3147365470080480, 23777750075552262

Offset: 0

Vladimir Kruchinin, Feb 07 2015

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A165817, A188687.

a := n -> (hypergeom([1/3, 2/3, -n], [1/2, 1], -27/4) +2 ) / 3:
seq(simplify(a(n)), n=0..20); # Peter Luschny, Feb 07 2015

FullSimplify[CoefficientList[Series[1 + x*D[Log[(2*Sin[(1/3)* ArcSin[(3/2)*Sqrt[3]* Sqrt[x/(1 - x)]]])/ (Sqrt[3]*Sqrt[(1 - x)* x])], x], {x, 0, 20}], x]] (* Vaclav Kotesovec, Feb 07 2015 *)
Table[(1 + Sum[Binomial[n, j]*Binomial[3*j-1, j], {j, 0, n}])/2, {n,0,20}] (* Vaclav Kotesovec, Feb 07 2015 after Vladimir Kruchinin *)

a(n):=(1+sum(binomial(n,j)*binomial(3*j-1,j),j,0,n))/2;

for(n=0,25, print1((1 + sum(k=0,n, binomial(n,k)*binomial(3*k-1,k)))/2, ", ")) \\ G. C. Greubel, Jun 03 2017

G.f.: x*G'(x)/G(x), where G(x) = x*(2/sqrt(3*x*(1-x)))*sin((1/3)*asin(3/2*sqrt(3*x/(1-x)))).
a(n) = (hypergeom([1/3,2/3,-n],[1/2,1],-27/4)+2)/3. - Peter Luschny, Feb 07 2015
From Vaclav Kotesovec, Feb 07 2015: (Start)
Recurrence: 2*n*(2*n-1)*a(n) = (43*n^2 - 53*n + 18)*a(n-1) - 3*(35*n^2 - 85*n + 54)*a(n-2) + (n-2)*(97*n - 165)*a(n-3) - 31*(n-3)*(n-2)*a(n-4).
a(n) ~ 31^(n+1/2) / (9 * sqrt(Pi*n) * 2^(2*n+1)).
(End)

A298610 Triangle read by rows, the unsigned coefficients of G(n, n, x/2) where G(n,a,x) denotes the n-th Gegenbauer polynomial, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 2, 0, 3, 0, 12, 0, 10, 10, 0, 60, 0, 35, 0, 105, 0, 280, 0, 126, 56, 0, 756, 0, 1260, 0, 462, 0, 840, 0, 4620, 0, 5544, 0, 1716, 330, 0, 7920, 0, 25740, 0, 24024, 0, 6435, 0, 6435, 0, 60060, 0, 135135, 0, 102960, 0, 24310

Offset: 0

Peter Luschny, Jan 25 2018

			[0]   1
[1]   0,    1
[2]   2,    0,    3
[3]   0,   12,    0,    10
[4]  10,    0,   60,     0,    35
[5]   0,  105,    0,   280,     0,    126
[6]  56,    0,  756,     0,  1260,      0,   462
[7]   0,  840,    0,  4620,     0,   5544,     0,   1716
[8] 330,    0, 7920,     0, 25740,      0, 24024,      0, 6435
[9]   0, 6435,    0, 60060,     0, 135135,     0, 102960,    0,  24310

T(2n, 0) = A165817(n). T(n,n) = A088218(n). Row sums are A213684.

Cf. A109187.

with(orthopoly):
seq(seq((-1)^iquo(n-k, 2)*coeff(G(n,n,x/2),x,k), k=0..n), n=0..9);

p[n_] := Binomial[3 n - 1, n] Hypergeometric2F1[-n, 3 n, n + 1/2, 1/2 - x/4];
Flatten[Table[(-1)^Floor[(n-k)/2] Coefficient[p[n], x, k], {n,0,9}, {k,0,n}]]

G(n, x) = binomial(3*n-1, n)*hypergeom([-n, 3*n], [n+1/2], 1/2 - x/4).

A371816 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(3n-3k-1,n-3*k).

Original entry on oeis.org

1, 2, 10, 55, 322, 1947, 12013, 75154, 474946, 3024742, 19381045, 124797862, 806875421, 5234713031, 34060165282, 222174355575, 1452425614146, 9513309908589, 62418283102246, 410161124310550, 2698932409666237, 17781425199962255, 117281204608676426

Offset: 0

Seiichi Manyama, Apr 06 2024

nonn

Cf. A120305, A165817, A371817.

Cf. A371770.

Table[Sum[(-1)^k Binomial[3n-3k-1,n-3k],{k,0,Floor[n/3]}],{n,0,30}] (* Harvey P. Dale, Aug 07 2025 *)

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

a(n) = [x^n] 1/((1+x^3) * (1-x)^(2*n)).

a(n) = binomial(3*n-1, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [1/3-n, 2/3-n, 1-n], -1). - Stefano Spezia, Apr 07 2024

A165984 Number of ways to put n indistinguishable balls into n^3 distinguishable boxes.

Original entry on oeis.org

1, 1, 36, 3654, 766480, 275234400, 151111164204, 117774526188844, 123672890985095232, 168324948170849366820, 288216356245328994082600, 606320062786763763996747618, 1537230010624231669678572481296, 4622745700243196227504110670860680

Offset: 0

Thomas Wieder, Oct 03 2009

nonn

See A165817 for the case n indistinguishable balls into 2*n distinguishable boxes.
See A054688 for the case n indistinguishable balls into n^2 distinguishable boxes.
a(n) is the number of (weak) compositions of n into n^3 parts. - Joerg Arndt, Oct 04 2017

			For n = 2 the a(2) = 36 solutions are
[0, 0, 0, 0, 0, 0, 0, 2]
[0, 0, 0, 0, 0, 0, 1, 1]
[0, 0, 0, 0, 0, 0, 2, 0]
[0, 0, 0, 0, 0, 1, 0, 1]
[0, 0, 0, 0, 0, 1, 1, 0]
[0, 0, 0, 0, 0, 2, 0, 0]
[0, 0, 0, 0, 1, 0, 0, 1]
[0, 0, 0, 0, 1, 0, 1, 0]
[0, 0, 0, 0, 1, 1, 0, 0]
[0, 0, 0, 0, 2, 0, 0, 0]
[0, 0, 0, 1, 0, 0, 0, 1]
[0, 0, 0, 1, 0, 0, 1, 0]
[0, 0, 0, 1, 0, 1, 0, 0]
[0, 0, 0, 1, 1, 0, 0, 0]
[0, 0, 0, 2, 0, 0, 0, 0]
[0, 0, 1, 0, 0, 0, 0, 1]
[0, 0, 1, 0, 0, 0, 1, 0]
[0, 0, 1, 0, 0, 1, 0, 0]
[0, 0, 1, 0, 1, 0, 0, 0]
[0, 0, 1, 1, 0, 0, 0, 0]
[0, 0, 2, 0, 0, 0, 0, 0]
[0, 1, 0, 0, 0, 0, 0, 1]
[0, 1, 0, 0, 0, 0, 1, 0]
[0, 1, 0, 0, 0, 1, 0, 0]
[0, 1, 0, 0, 1, 0, 0, 0]
[0, 1, 0, 1, 0, 0, 0, 0]
[0, 1, 1, 0, 0, 0, 0, 0]
[0, 2, 0, 0, 0, 0, 0, 0]
[1, 0, 0, 0, 0, 0, 0, 1]
[1, 0, 0, 0, 0, 0, 1, 0]
[1, 0, 0, 0, 0, 1, 0, 0]
[1, 0, 0, 0, 1, 0, 0, 0]
[1, 0, 0, 1, 0, 0, 0, 0]
[1, 0, 1, 0, 0, 0, 0, 0]
[1, 1, 0, 0, 0, 0, 0, 0]
[2, 0, 0, 0, 0, 0, 0, 0]

Cf. A001700, A054688, A060690, A165817.

a:= n-> binomial(n^3+n-1, n): seq(a(n), n=0..16);

Table[Binomial[n^3 + n - 1, n], {n, 0, 13}] (* Michael De Vlieger, Oct 05 2017 *)

a(n) = binomial(n^3+n-1, n); \\ Altug Alkan, Oct 03 2017

a(n) = binomial(n^3+n-1, n).
Let denote P(n) = the number of integer partitions of n,
p(i) = the number of parts of the i-th partition of n,
d(i) = the number of different parts of the i-th partition of n,
m(i,j) = multiplicity of the j-th part of the i-th partition of n.
Then one has:
a(n) = Sum_{i=1..P(n)} (n^3)!/((n^3-p(i))!*(Product_{j=1..d(i)} m(i,j)!)).
a(n) = [x^n] 1/(1 - x)^(n^3). - Ilya Gutkovskiy, Oct 03 2017

cp's OEIS Frontend

A386006 a(n) = Sum_{k=0..n} binomial(3*n-2,k).

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A323324 Coefficients T(n,k) of x^n*y^(n-k)*z^k in function A = A(x,y,z) such that A = 1 + x*B*C, B = 1 + y*C*A, and C = 1 + z*A*B, as a triangle read by rows.

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A348478 Number of compositions of n into exactly n nonnegative parts such that each positive i-th part has the same parity as i.

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A365771 a(n) = binomial(2*n+1, n)/(2*n+1) * binomial(3*n-1, n) for n >= 0.

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A373865 Sum over all compositions of 2n into n parts of the least common multiple of the parts.

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A254747 a(n) = (1 + Sum_{j=0..n} (C(n,j)*C(3*j-1,j))) / 2.

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A298610 Triangle read by rows, the unsigned coefficients of G(n, n, x/2) where G(n,a,x) denotes the n-th Gegenbauer polynomial, T(n, k) for 0 <= k <= n.

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A371816 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(3*n-3*k-1,n-3*k).

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A323324 Coefficients T(n,k) of x^ny^(n-k)z^k in function A = A(x,y,z) such that A = 1 + xBC, B = 1 + yCA, and C = 1 + zAB, as a triangle read by rows.

A365771 a(n) = binomial(2n+1, n)/(2n+1) * binomial(3*n-1, n) for n >= 0.

A254747 a(n) = (1 + Sum_{j=0..n} (C(n,j)C(3j-1,j))) / 2.

A371816 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(3n-3k-1,n-3*k).