A165817 -id:A165817 - cp's OEIS frontend

A350383 a(n) = [x^n] 1/(1 + x + x^2)^n.

1, -1, 1, 2, -15, 49, -98, 48, 561, -2860, 8151, -12948, -9282, 149226, -594320, 1428952, -1448655, -5538975, 37450900, -122995950, 239589735, -37528755, -1886983020, 8939152560, -24579514050, 35197176924, 51580335366, -541312482256, 2033695030128, -4624358661240

Offset: 0

Seiichi Manyama, Dec 29 2021

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A002426, A027907, A103779, A110556, A165817, A213684, A228960, A350406, A350407, A362087.

a := n -> (-1)^n*hypergeom([-n/3, 1/3 - n/3, 2/3 - n/3, n], [1/3, 2/3, 1], 1): seq(simplify(a(n)), n = 0..30); # Peter Bala, Apr 17 2023

a[n_] := Coefficient[Series[1/(1 + x + x^2)^n, {x, 0, n}], x, n]; Array[a, 30, 0] (* Amiram Eldar, Dec 29 2021 *)

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

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1+k,k) * binomial(n,3*k).
Recurrence: 3*(n-1)*n*(4*n - 7)*a(n) = -2*(n-1)*(28*n^2 - 63*n + 27)*a(n-1) - 3*(3*n - 5)*(3*n - 4)*(4*n - 3)*a(n-2). - Vaclav Kotesovec, Mar 18 2023
From Peter Bala, Apr 15 2023: (Start)
a(n) = (-1)^n*hypergeom([-n/3, 1/3 - n/3, 2/3 - n/3, n], [1/3, 2/3, 1], 1).
Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for positive integers n and r and all primes p >= 5. Cf. A228960.
More generally, let k be a positive integer, m an integer and let f(x) = g(x)/h(x), where g(x) and h(x) are both finite products of cyclotomic polynomials. Then we conjecture that the same supercongruences hold, except for a finite number of primes p depending on f(x), for the sequence {a_(k,m,f)(n): n >= 0} defined by a_(k,m,f)(n) = [x^(k*n)] f(x)^(m*n). (End)
From Peter Bala, Mar 11 2025: (Start)
G.f.: A(x) = 1 + x*d/dx(log(G(x)/x)), where G(x) = x - x^2 + x^3 - 4*x^5 + 14*x^6 - 30*x^7 + ... is the g.f. of A103779.
The following formulas hold for n >= 1:
a(n) = [x^n] T(2*n, (1 - x)/2), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind.
a(n) = Sum_{k = 0..n} (-1)^(n+k) * n/(2*n-k) * binomial(2*n-k, k)*binomial(2*n-2*k, n).
a(n) = (1/2)*(-1)^n*binomial(2*n, n)*hypergeom([-n/2, (-n+1)/2], [-2*n+1], 4). Cf. A213684. (End)

A352373 a(n) = [x^n] ( 1/((1 - x)^2*(1 - x^2)) )^n for n >= 1.

Original entry on oeis.org

2, 12, 74, 484, 3252, 22260, 154352, 1080612, 7621526, 54071512, 385454940, 2758690636, 19810063392, 142662737376, 1029931873824, 7451492628260, 54013574117106, 392188079586468, 2851934621212598, 20766924805302984, 151403389181347160, 1105047483656041080

Offset: 1

Peter Bala, Mar 14 2022

Suppose n identical objects are distributed in 3*n labeled baskets, 2*n colored white and n colored black. White baskets can contain any number of objects (or be empty), while black baskets must contain an even number of objects (or be empty). a(n) is the number of distinct possible distributions.
Number of nonnegative integer solutions to n = x_1 + x_2 + ... + x_(2*n) + 2*y_1 + 2*y_2 + ... + 2*y_n.
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.
Calculation suggests that, in fact, stronger congruences may hold.
Conjecture: the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and positive integers n and k.
More generally, let r and s be integers and define a sequence (a(r,s;n))n>=1 by a(r,s;n) = [x^n] ( (1 + x)^r * (1 - x)^s )^n.
Conjecture: for each r and s the above supercongruences hold for the sequence (a(r,s;n))n>=1.
The present sequence is the case r = -1 and s = -3. Other cases include A000984 (r = 2, s = 0), A001700 with offset 1 (r = 0, s = -1), A002003 (r = 1, s = -1), A091527 (r = 3, s = -1), A119259 (r = 2, s = -1), A156894 (r = 1, s = -2), A165817 (r = 0, s = -2), A234839 (r = 1, s = 2), A348410 (r = -1, s = -2) and A351857 (r = -2, s = -4).

			n = 2: 12 distributions of 2 identical objects in 4 white and 2 black baskets
             White         Black
   1)   (0) (0) (0) (0)   [2] [0]
   2)   (0) (0) (0) (0)   [0] [2]
   3)   (2) (0) (0) (0)   [0] [0]
   4)   (0) (2) (0) (0)   [0] [0]
   5)   (0) (0) (2) (0)   [0] [0]
   6)   (0) (0) (0) (2)   [0] [0]
   7)   (1) (1) (0) (0)   [0] [0]
   8)   (1) (0) (1) (0)   [0] [0]
   9)   (1) (0) (0) (1)   [0] [0]
  10)   (0) (1) (1) (0)   [0] [0]
  11)   (0) (1) (0) (1)   [0] [0]
  12)   (0) (0) (1) (1)   [0] [0]
Examples of supercongruences:
a(7) - a(1) = 154352 - 2 = 2*(3^2)*(5^2)*(7^3) == 0 (mod 7^3);
a(2*11) - a(2) = 1105047483656041080 - 12 = (2^2)*3*(11^3)*13*101*103*2441* 209581 == 0 (mod 11^3).

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

Paolo Xausa, Table of n, a(n) for n = 1..1000

Cf. A000984, A001448, A001700, A002003, A091527, A119259, A156894, A165817, A211419, A211421, A234839, A262733, A276098, A348410, A351856, A351857.

seq(add( binomial(3*n-2*k-1,n-2*k)*binomial(n+k-1,k), k = 0..floor(n/2)), n = 1..25);

nterms=25;Table[Sum[Binomial[3n-2k-1,n-2k]Binomial[n+k-1,k],{k,0,Floor[n/2]}],{n,nterms}] (* Paolo Xausa, Apr 10 2022 *)

a(n) = Sum_{k = 0..floor(n/2)} binomial(3*n-2*k-1,n-2*k)*binomial(n+k-1,k).
a(n) = Sum_{k = 0..n} (-1)^k*binomial(4*n-k-1,n-k)*binomial(n+k-1,k).
a(n) = binomial(4*n-1,n)*hypergeom([n, -n], [1-4*n], -1).
48*n*(n-1)*(3*n-1)*(3*n-2)*(93*n^3-434*n^2+668*n-339)*a(n) = 12*(n-1)*(21762*n^6-134199*n^5+323805*n^4-386685*n^3+237728*n^2-70336*n+7680)*a(n-1) + 5*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)*(93*n^3-155*n^2+79*n-12)*a(n-2) with a(1) = 2 and a(2) = 12.
The o.g.f. A(x) = 2*x + 12*x^2 + 74*x^3 + ... is the diagonal of the bivariate rational function x*t/(1 - t/((1 - x)^2*(1 - x^2))) and hence is an algebraic function over Q(x) by Stanley 1999, Theorem 6.33, p. 197.
A(x) = x*d/dx(log(F(x))), where F(x) = (1/x)*Series_Reversion( x*(1 - x)^2*(1 - x^2) ).
a(n) ~ sqrt(4 + sqrt(6)) * (13/4 + 31*sqrt(6)/18)^n / (2*sqrt(5*Pi*n)). - Vaclav Kotesovec, Mar 15 2022

A371770 a(n) = Sum_{k=0..floor(n/3)} binomial(3n-3k-1,n-3*k).

Original entry on oeis.org

1, 2, 10, 57, 338, 2057, 12741, 79914, 505954, 3226638, 20696685, 133382658, 862978221, 5601919325, 36467212610, 237974911737, 1556281907586, 10196788555859, 66921360130374, 439860632463462, 2895002186799453, 19077000179746293, 125849150650146714

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Robert Israel, Table of n, a(n) for n = 0..1202

Cf. A005809, A165817, A183160.

Cf. A371758, A371771, A371772.

f:= proc(n) local k; add(binomial(3*n-3*k-1,n-3*k),k=0..n/3) end proc:
map(f, [$0..30]); # Robert Israel, Feb 28 2025

a(n) = sum(k=0, n\3, 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 06 2024
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: 18*n*(2*n - 1)*(13*n - 22)*(37*n - 51)*a(n) = 3*(40885*n^4 - 165468*n^3 + 229373*n^2 - 125562*n + 22680)*a(n-1) - (40885*n^4 - 165468*n^3 + 229373*n^2 - 125562*n + 22680)*a(n-2) + 3*(3*n - 5)*(3*n - 4)*(13*n - 9)*(37*n - 14)*a(n-3).
a(n) ~ 3^(3*n + 5/2) / (13 * sqrt(Pi*n) * 2^(2*n+1)). (End)

A385004 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n,k).

Original entry on oeis.org

1, 5, 31, 200, 1311, 8665, 57556, 383556, 2561871, 17140007, 114819351, 769925568, 5166845124, 34696155564, 233113911208, 1566926561740, 10536427052463, 70872688450083, 476854924775869, 3209222876463192, 21602639249766951, 145444151677134153, 979397744169608784

Offset: 0

Seiichi Manyama, Jul 30 2025

Cf. A000244, A100192, A385498, A386699.
Cf. A005809, A066380, A165817, A371813, A386700.
Cf. A001764, A386617.

Table[(27/4)^n - Binomial[3*n, n] * (-1 + Hypergeometric2F1[1, -2*n, 1 + n, -1/2]), {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)

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

a(n) = [x^n] 1/((1-3*x) * (1-x)^(2*n)).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(2*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 8*n*(2*n - 1)*(15*n - 23)*a(n) = 6*(540*n^3 - 1503*n^2 + 1239*n - 320)*a(n-1) - 81*(3*n - 5)*(3*n - 4)*(15*n - 8)*a(n-2).
a(n) ~ 3^(3*n) / 2^(2*n+1) * (1 + 5/(3*sqrt(3*Pi*n))). (End)
G.f.: g/(3-2*g)^2 where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^2*(9-4*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 17 2025

A386700 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(3*n,k).

Original entry on oeis.org

1, 0, 6, 30, 186, 1140, 7116, 44856, 285066, 1823232, 11721726, 75683718, 490429224, 3187723344, 20774505408, 135699314640, 888177411018, 5823660624408, 38245666664994, 251528316024042, 1656338630258826, 10919849458481028, 72068276593960884, 476093333668519872

Offset: 0

Seiichi Manyama, Jul 30 2025

Cf. A122803, A386701, A386702.

Cf. A005809, A066380, A165817, A371813, A385004.

Table[(-8/9)^n - Binomial[3*n, n]*(-1 + Hypergeometric2F1[1, -2*n, 1 + n, 1/3]), {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)

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

a(n) = [x^n] 1/((1+2*x) * (1-x)^(2*n)).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(2*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 18*n*(2*n - 1)*(55*n^2 - 175*n + 138)*a(n) = (11605*n^4 - 49410*n^3 + 74243*n^2 - 46014*n + 9720)*a(n-1) + 24*(3*n - 5)*(3*n - 4)*(55*n^2 - 65*n + 18)*a(n-2).
a(n) ~ 3^(3*n + 1/2) / (5 * sqrt(Pi*n) * 2^(2*n)). (End)
G.f.: g/((-2+3*g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - 6*x*g^2*(-1+g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 17 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

A348476 Number of compositions of n into exactly n nonnegative parts such that all positive parts are odd.

Original entry on oeis.org

1, 1, 1, 4, 13, 36, 106, 323, 981, 2992, 9196, 28392, 87946, 273287, 851579, 2659764, 8324357, 26100560, 81969496, 257800532, 811862268, 2559731360, 8079294664, 25525787344, 80719066698, 255466082911, 809138591431, 2564605664428, 8134003910311, 25813957574292

Offset: 0

Alois P. Heinz, Oct 19 2021

nonn

			a(0) = 1: [].
a(1) = 1: [1].
a(2) = 1: [1,1].
a(3) = 4: [3,0,0], [0,3,0], [0,0,3], [1,1,1].
a(4) = 13: [3,1,0,0], [3,0,1,0], [3,0,0,1], [1,3,0,0], [0,3,1,0], [0,3,0,1],[1,0,3,0], [0,1,3,0], [0,0,3,1], [1,0,0,3], [0,1,0,3], [0,0,1,3], [1,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..1968

Cf. A001700, A088218, A165817, A305161, A324969.

b:= proc(n, t) option remember; `if`(t=0, 1-signum(n),
      add(`if`(j=0 or j::odd, b(n-j, t-1), 0), j=0..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);

b[n_, t_] := b[n, t] = If[t == 0, 1 - Sign[n],
     Sum[If[j == 0 || OddQ[j], b[n - j, t - 1], 0], {j, 0, n}]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 16 2022, after Alois P. Heinz *)

a(n) ~ c * d^n / sqrt(Pi*n), where d = 3.22870495109450172934784925586... is largest positive root of the equation 4*d^4 - 12*d^3 + 4*d^2 - 24*d + 5 = 0 and c = 0.4302331663731241127284415754... is positive root of the equation 5824*c^8 - 32*c^4 - 4*c^2 - 5 = 0. - Vaclav Kotesovec, Nov 01 2021

A349966 a(n) = Sum_{k=0..n} (k * (n-k))^n.

Original entry on oeis.org

1, 0, 1, 16, 418, 17600, 1086979, 92223488, 10292241540, 1462309109760, 257739952352133, 55188518041440256, 14111052911099343782, 4246668467339066589184, 1485904567816768099571207, 598145009954138900489830400

Offset: 0

Seiichi Manyama, Dec 07 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..240

Cf. A033455, A100262, A145216, A165817, A306548, A349964, A349965.

a[0] = 1; a[n_] := Sum[(k*(n - k))^n, {k, 0, n}]; Array[a, 16, 0] (* Amiram Eldar, Dec 07 2021 *)

PARI
```
a(n) = sum(k=0, n, (k*(n-k))^n);
```

a(n) = [x^n] (Sum_{k=0..n} k^n * x^k)^2.

a(n) ~ sqrt(Pi) * n^(2*n + 1/2) / 2^(2*n + 1). - Vaclav Kotesovec, Dec 07 2021

A371813 a(n) = Sum_{k=0..n} (-1)^k * binomial(3*n-k-1,n-k).

Original entry on oeis.org

1, 1, 7, 40, 239, 1461, 9076, 57044, 361711, 2309467, 14827487, 95630272, 619111172, 4021011580, 26187682024, 170960159100, 1118406332655, 7330011083079, 48119501497909, 316354663355384, 2082573599282359, 13726029056757029, 90565080767425744

Offset: 0

Seiichi Manyama, Apr 06 2024

nonn

Cf. A072547, A371814.

Cf. A005809, A066380, A165817, A385004, A386700.

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

a(n) = [x^n] 1/((1+x) * (1-x)^(2*n)).
a(n) = binomial(3*n-1, n)*hypergeom([1, -n], [1-3*n], -1). - Stefano Spezia, Apr 07 2024
From Vaclav Kotesovec, Apr 07 2024: (Start)
Recurrence: 8*n*(2*n - 1)*(28*n^2 - 87*n + 67)*a(n) = 2*(1456*n^4 - 6008*n^3 + 8593*n^2 - 4949*n + 960)*a(n-1) + 3*(3*n - 5)*(3*n - 4)*(28*n^2 - 31*n + 8)*a(n-2).
a(n) ~ 3^(3*n + 1/2) / (sqrt(Pi*n) * 2^(2*n+2)). (End)
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n,k). - Seiichi Manyama, Jul 30 2025
G.f.: g/((-1+2*g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^2*(-3+4*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 17 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

A350383 a(n) = [x^n] 1/(1 + x + x^2)^n.

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A352373 a(n) = [x^n] ( 1/((1 - x)^2*(1 - x^2)) )^n for n >= 1.

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A371770 a(n) = Sum_{k=0..floor(n/3)} binomial(3*n-3*k-1,n-3*k).

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A385004 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n,k).

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A386700 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(3*n,k).

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

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A348476 Number of compositions of n into exactly n nonnegative parts such that all positive parts are odd.

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A349966 a(n) = Sum_{k=0..n} (k * (n-k))^n.

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A371770 a(n) = Sum_{k=0..floor(n/3)} binomial(3n-3k-1,n-3*k).