A344560 -id:A344560 - cp's OEIS frontend

A361637 Constant term in the expansion of (1 + x + y + z + 1/(xyz))^n.

1, 1, 1, 1, 25, 121, 361, 841, 4201, 25705, 118441, 423721, 1628881, 8065201, 41225185, 184416961, 768211081, 3420474121, 16620237001, 79922011465, 364149052705, 1638806098945, 7655390077105, 36739991161105, 174363209490625, 811840219629121, 3790118889635521

Offset: 0

Seiichi Manyama, Mar 19 2023

nonn

Diagonal of the rational function 1/(1 - (x^4 + y^4 + z^4 + w^4 + x*y*z*w)). - Seiichi Manyama, Jul 04 2025

Winston de Greef, Table of n, a(n) for n = 0..1429

Cf. A002426, A344560, A361703.
Cf. A008977, A328725.
Cf. A082488, A274783, A274785.

a(n) = n!*sum(k=0, n\4, 1/(k!^4*(n-4*k)!));

a(n) = n! * Sum_{k=0..floor(n/4)} 1/(k!^4 * (n-4*k)!).
G.f.: Sum_{k>=0} (4*k)!/k!^4 * x^(4*k)/(1-x)^(4*k+1).
From Vaclav Kotesovec, Mar 20 2023: (Start)
Recurrence: n^3*a(n) = (2*n - 1)*(2*n^2 - 2*n + 1)*a(n-1) - (n-1)*(6*n^2 - 12*n + 7)*a(n-2) + 2*(n-2)*(n-1)*(2*n - 3)*a(n-3) + 255*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 5^(n + 3/2) / (2^(5/2) * Pi^(3/2) * n^(3/2)). (End)

A208425 Expansion of Sum_{n>=0} (3n)!/n!^3 x^(2n)/(1-x)^(3n+1).

Original entry on oeis.org

1, 1, 7, 25, 151, 751, 4411, 24697, 146455, 862351, 5195257, 31392967, 191815339, 1177508515, 7276161907, 45154764025, 281492498455, 1761076827895, 11055132835705, 69600761349175, 439370198255401, 2780265190892641, 17631718101804517, 112038660509078695

Offset: 0

Paul D. Hanna, Feb 26 2012

nonn

Compare g.f. to: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)/(1-2*x)^(3*n+1), which is a g.f. of the Franel numbers (A000172).
From Zhi-Wei Sun, Nov 12 2016: (Start)
Conjecture: (i) For any prime p > 3 and positive integer n, the number (a(p*n)-a(n))/(p*n)^3 is always a p-adic integer.
(ii) For any prime p == 1 (mod 3), we have Sum_{k=0..p-1}a(k) == C(2(p-1)/3,(p-1)/3) (mod p^2). For any prime p == 2 (mod 3), we have Sum_{k=0..p-1}a(k) == 2p/C(2(p+1)/3,(p+1)/3) (mod p^2).
We have proved part (i) of this conjecture for n = 1. (End)
Diagonal of rational functions 1/(1 - x*y - y*z - x*z - x*y*z), 1/(1 - x*y + y*z + x*z - x*y*z). - Gheorghe Coserea, Jul 03 2018
Number of paths from (0,0,0) to (n,n,n) using steps (1,1,0), (1,0,1), (0,1,1), and (1,1,1). - William J. Wang, Dec 07 2020
Diagonal of the rational function 1/(1 - (x^2 + y^2 + z^2 + x*y*z)). - Seiichi Manyama, Jul 04 2025

			G.f.: A(x) = 1 + x + 7*x^2 + 25*x^3 + 151*x^4 + 751*x^5 + 4411*x^6 +...
where
A(x) = 1/(1-x) + 6*x^2/(1-x)^4 + 90*x^4/(1-x)^7 + 1680*x^6/(1-x)^10 + 34650*x^8/(1-x)^13 + 756756*x^10/(1-x)^16 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
A. Bostan, S. Boukraa, J.-M. Maillard and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
Hao Pan and Zhi-Wei Sun, Supercongruences for central trinomial coefficients, arXiv:2012.05121 [math.NT], 2020.
Zhi-Wei Sun, Supercongruences involving Lucas sequences, arXiv:1610.03384 [math.NT], 2016.

Cf. A000172, A001850, A208426, A244973, A274783.

Cf. A081798, A344560.

series(hypergeom([1/3, 2/3], [1], 27*x^2/(1 - x)^3)/(1 - x), x=0, 25): seq(coeff(%, x, n), n=0..23);  # Mark van Hoeij, May 20 2013
a := n -> hypergeom([1/2 - n/2, -n/2, n + 1], [1, 1], 4); seq(simplify(a(n)), n=0..23);  # Peter Luschny, Jan 11 2025

nmax = 20; CoefficientList[Series[Sum[(3*n)!/n!^3 * x^(2*n)/(1-x)^(3*n+1), {n, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 05 2016 *)

{a(n)=polcoeff(sum(m=0,n, (3*m)!/m!^3*x^(2*m)/(1-x+x*O(x^n))^(3*m+1)),n)}
for(n=0,25,print1(a(n),", "))

Conjecture: n^2*(3*n-5)*a(n) +(-9*n^3+24*n^2-17*n+4) *a(n-1) -(3*n-4) *(24*n^2-56*n+27)*a(n-2) -(3*n-2)*(n-2)^2*a(n-3)=0. - R. J. Mathar, Mar 10 2016
a(n) ~ sqrt(1/2 + sqrt(13)*cos(arctan(53*sqrt(3)/19)/3)/6) * (1 + 6*cos(Pi/9))^n / (Pi*n). - Vaclav Kotesovec, Jul 05 2016
It is easy to show that a(n) = Sum_{k=0..n}C(n,k)*C(n-k,k)*C(n+k,k) = Sum_{k=0..n}C(n+k,k)*C(n,2k)*C(2k,k). By this formula and the Zeilberger algorithm, we confirm the recurrence conjectured by R. J. Mathar. - Zhi-Wei Sun, Nov 12 2016
G.f. y=A(x) satisfies: 0 = x*(x + 2)*(x^3 + 24*x^2 + 3*x - 1)*y'' + (3*x^4 + 56*x^3 + 147*x^2 + 12*x - 2)*y' + (x^3 + 9*x^2 + 42*x + 2)*y. - Gheorghe Coserea, Jul 03 2018
a(n) = hypergeom([1/2 - n/2, -n/2, n + 1], [1, 1], 4). - Peter Luschny, Jan 11 2025

A361703 Constant term in the expansion of (1 + w + x + y + z + 1/(wxy*z))^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 121, 721, 2521, 6721, 15121, 143641, 1302841, 7579441, 32586841, 113753641, 509068561, 3599319361, 25076993761, 142188273361, 662296228561, 2933770097881, 15581813723281, 99333170493481, 623696622059281, 3466773281312881, 17406784944114721

Offset: 0

Seiichi Manyama, Mar 21 2023

Diagonal of the rational function 1/(1 - (v^5 + w^5 + x^5 + y^5 + z^5 + v*w*x*y*z)). - Seiichi Manyama, Jul 04 2025

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

Cf. A361675, A361704, A361705.
Cf. A002426, A344560, A361637.
Cf. A082489, A361636.

a(n) = sum(k=0, n\5, (4*k)!/k!^4*binomial(5*k, 4*k)*binomial(n, 5*k));

a(n) = Sum_{k=0..floor(n/5)} (4*k)!/k!^4 * binomial(5*k,4*k) * binomial(n,5*k) = Sum_{k=0..floor(n/5)} (5*k)!/k!^5 * binomial(n,5*k).

a(n) ~ 9 * 6^n / (sqrt(5) * Pi^2 * n^2). - Vaclav Kotesovec, Mar 25 2023

A361675 Constant term in the expansion of (1 + xyz + wyz + wxz + wxy + 1/(wxy*z))^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 841, 6721, 30241, 100801, 277201, 665281, 1441441, 10450441, 118918801, 917716801, 5162277121, 23183465761, 88037913601, 293383742401, 988690080001, 4810025534161, 33669381872281, 234722545854721, 1407984124932001, 7219196588604001

Offset: 0

Seiichi Manyama, Mar 20 2023

nonn

Also constant term in the expansion of (1 + w^3 + x^3 + y^3 + z^3 + 1/(w*x*y*z))^n.

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

Cf. A344560, A361673.

Table[n! * Sum[1/(k!^4 * (3*k)! * (n-7*k)!), {k,0,n/7}], {n,0,30}] (* Vaclav Kotesovec, Mar 22 2023 *)

a(n) = n!*sum(k=0, n\7, 1/(k!^4*(3*k)!*(n-7*k)!));

a(n) = n! * Sum_{k=0..floor(n/7)} 1/(k!^4 * (3*k)! * (n-7*k)!) = Sum_{k=0..floor(n/7)} (4*k)!/k!^4 * binomial(7*k,4*k) * binomial(n,7*k).
From Vaclav Kotesovec, Mar 22 2023: (Start)
Recurrence: 3*n^4*(3*n - 14)*(3*n - 7)*a(n) = 3*(63*n^6 - 567*n^5 + 1750*n^4 - 2555*n^3 + 2114*n^2 - 931*n + 170)*a(n-1) - 21*(n-1)*(27*n^5 - 270*n^4 + 995*n^3 - 1770*n^2 + 1579*n - 570)*a(n-2) + 21*(n-2)*(n-1)*(45*n^4 - 450*n^3 + 1625*n^2 - 2580*n + 1547)*a(n-3) - 105*(n-3)*(n-2)*(n-1)*(9*n^3 - 81*n^2 + 239*n - 235)*a(n-4) + 21*(n-4)*(n-3)*(n-2)*(n-1)*(27*n^2 - 189*n + 329)*a(n-5) - 189*(n-5)*(n-4)^2*(n-3)*(n-2)*(n-1)*a(n-6) + 823570*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-7).
a(n) ~ sqrt(c) * (1 + 7/3^(3/7))^n / (Pi^2 * n^2), where c = 16.2900695424464373693361847496482396571795561541696471874653361... is the real root of the equation -28752928904042094750625 + 28055343229566040503381*c - 11938039301954303025264*c^2 + 2737803069771369110784*c^3 - 386503281377426239488*c^4 + 32401195469663698944*c^5 - 1511492100446748672*c^6 + 30217251487481856*c^7 = 0. (End)

A361657 Constant term in the expansion of (1 + x^2 + y^2 + 1/(x*y))^n.

Original entry on oeis.org

1, 1, 1, 1, 13, 61, 181, 421, 1261, 5293, 21421, 73261, 232321, 789361, 2954953, 11127481, 39961741, 139908301, 499315501, 1835933293, 6792310153, 24827506873, 90058277233, 328509505633, 1210097040769, 4473191880961, 16495696956961, 60721903812961

Offset: 0

Seiichi Manyama, Mar 19 2023

nonn

Cf. A002426, A361658.

Cf. A000897, A344560, A361637.

Table[n!*Sum[1/(k!^2*(2*k)!*(n - 4*k)!), {k, 0, n/4}], {n, 0, 30}] (* Vaclav Kotesovec, Mar 20 2023 *)

a(n) = n!*sum(k=0, n\4, 1/(k!^2*(2*k)!*(n-4*k)!));

a(n) = n! * Sum_{k=0..floor(n/4)} 1/(k!^2 * (2*k)! * (n-4*k)!) = Sum_{k=0..floor(n/4)} binomial(n,4*k) * A000897(k).
From Vaclav Kotesovec, Mar 20 2023: (Start)
Recurrence: (n-2)*n^2*a(n) = (4*n^3 - 12*n^2 + 10*n - 3)*a(n-1) - (n-1)*(6*n^2 - 18*n + 13)*a(n-2) + 4*(n-2)^2*(n-1)*a(n-3) + 63*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ (1 + 2*sqrt(2))^(n+1) / (4*Pi*n). (End)

A361673 Constant term in the expansion of (1 + xy + yz + zx + 1/(xy*z))^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 61, 361, 1261, 3361, 7561, 34021, 235621, 1294921, 5482621, 19039021, 65345281, 286147681, 1511480881, 7688794681, 34337600281, 138221512741, 554603041441, 2454508134541, 11874549049441, 57412094595241, 261925516443361, 1134301869703861

Offset: 0

Seiichi Manyama, Mar 20 2023

nonn

Also constant term in the expansion of (1 + x^2 + y^2 + z^2 + 1/(x*y*z))^n.

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

Cf. A344560, A361675.
Cf. A361637, A361658.
Cf. A001460.

Table[n! * Sum[1/(k!^3 * (2*k)! * (n-5*k)!), {k,0,n/5}], {n,0,20}] (* Vaclav Kotesovec, Mar 22 2023 *)

a(n) = n!*sum(k=0, n\5, 1/(k!^3*(2*k)!*(n-5*k)!));

a(n) = n! * Sum_{k=0..floor(n/5)} 1/(k!^3 * (2*k)! * (n-5*k)!) = Sum_{k=0..floor(n/5)} binomial(n,5*k) * A001460(k).
From Vaclav Kotesovec, Mar 22 2023: (Start)
Recurrence: 2*n^3*(2*n - 5)*a(n) = 2*(10*n^4 - 40*n^3 + 50*n^2 - 30*n + 7)*a(n-1) - 10*(n-1)*(4*n^3 - 18*n^2 + 26*n - 13)*a(n-2) + 40*(n-2)^3*(n-1)*a(n-3) - 10*(n-3)*(n-2)*(n-1)*(2*n - 5)*a(n-4) + 3129*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ sqrt(c) * (1 + 5/2^(2/5))^n / (Pi^(3/2) * n^(3/2)), where c = 3.154712586460560795509193778252140601572145506226776094640234924884123818... is the real root of the equation -30634915689 + 95407210000*c - 127160000000*c^2 + 79846400000*c^3 - 25600000000*c^4 + 3276800000*c^5 = 0. (End)

A361677 Constant term in the expansion of (1 + x + y + z + 1/(xy) + 1/(yz) + 1/(z*x))^n.

Original entry on oeis.org

1, 1, 1, 19, 73, 181, 1711, 10081, 38809, 256033, 1696861, 8388271, 49449511, 326195299, 1847392093, 10789655059, 69202030969, 418647580489, 2498113460881, 15735859252147, 97919649290053, 598317173139313, 3748943081117323

Offset: 0

Seiichi Manyama, Mar 20 2023

nonn

Cf. A201805, A361678.

Cf. A275047, A344560.

Table[Sum[(3*k)!/k!^3 * Binomial[3*k,k] * Binomial[n,3*k], {k,0,n/3}], {n,0,25}] (* Vaclav Kotesovec, Mar 22 2023 *)

a(n) = sum(k=0, n\3, (3*k)!/k!^3*binomial(3*k, k)*binomial(n, 3*k));

a(n) = Sum_{k=0..floor(n/3)} (3*k)!/k!^3 * binomial(3*k,k) * binomial(n,3*k).
From Vaclav Kotesovec, Mar 22 2023: (Start)
Recurrence: 2*n^3*(2*n - 3)*a(n) = 2*(10*n^4 - 32*n^3 + 38*n^2 - 22*n + 5)*a(n-1) - 2*(n-1)*(2*n - 3)*(10*n^2 - 24*n + 17)*a(n-2) + (n-2)*(n-1)*(769*n^2 - 2331*n + 1594)*a(n-3) - 2*(n-3)*(n-2)*(n-1)*(739*n - 1481)*a(n-4) + 733*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ sqrt(733/108 + 1/2^(2/3) + 9/2^(4/3)) * (1 + 9/2^(2/3))^n / (2 * Pi^(3/2) * n^(3/2)). (End)

A361699 Constant term in the expansion of (1 + x^3 + y^3 + 1/(x*y))^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 21, 121, 421, 1121, 2521, 6301, 23101, 99001, 386101, 1301301, 3943941, 11779041, 38241841, 136988041, 504616441, 1793870941, 6061831441, 19923689941, 66139128441, 227052188441, 800641330721, 2831644750221, 9870443816221, 33869987735221

Offset: 0

Seiichi Manyama, Mar 21 2023

Winston de Greef, Table of n, a(n) for n = 0..1798

Cf. A344560, A361657, A361700.

Table[Sum[Binomial[2*k,k] * Binomial[5*k,2*k] * Binomial[n,5*k], {k,0,n/5}], {n,0,20}] (* Vaclav Kotesovec, Mar 22 2023 *)

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

a(n) = Sum_{k=0..floor(n/5)} binomial(2*k,k) * binomial(5*k,2*k) * binomial(n,5*k).
From Vaclav Kotesovec, Mar 22 2023: (Start)
Recurrence: 3*n^2*(3*n - 10)*(3*n - 5)*a(n) = 3*(45*n^4 - 270*n^3 + 510*n^2 - 375*n + 104)*a(n-1) - 45*(n-1)*(6*n^3 - 36*n^2 + 67*n - 40)*a(n-2) + 15*(n-2)*(n-1)*(18*n^2 - 90*n + 109)*a(n-3) - 135*(n-3)^2*(n-2)*(n-1)*a(n-4) + 3152*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ sqrt(c) * (1 + 5/3^(3/5))^n / (Pi * n), where c = 0.8011502211360696582191471740430432783906089377204901279920664641344364478... is the real root of the equation -2483776 + 28284375*c - 141840000*c^2 + 337500000*c^3 - 405000000*c^4 + 194400000*c^5 = 0. (End)

A361700 Constant term in the expansion of (1 + x^4 + y^4 + 1/(x*y))^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 31, 211, 841, 2521, 6301, 13861, 30691, 90091, 360361, 1501501, 5645641, 18749641, 56063281, 157520641, 445836901, 1368402421, 4638690211, 16511900791, 58059667051, 195211574251, 625463703151, 1942351017751, 6016826006101, 19113287111101

Offset: 0

Seiichi Manyama, Mar 21 2023

Winston de Greef, Table of n, a(n) for n = 0..1886

Cf. A344560, A361657, A361699.

Table[Sum[Binomial[2*k,k] * Binomial[6*k,2*k] * Binomial[n,6*k], {k,0,n/6}], {n,0,20}] (* Vaclav Kotesovec, Mar 22 2023 *)

a(n) = sum(k=0, n\6, binomial(2*k, k)*binomial(6*k, 2*k)*binomial(n, 6*k));

a(n) = Sum_{k=0..floor(n/6)} binomial(2*k,k) * binomial(6*k,2*k) * binomial(n,6*k).
From Vaclav Kotesovec, Mar 22 2023: (Start)
Recurrence: (n-3)*n^2*(2*n - 9)*(2*n - 3)*a(n) = (24*n^5 - 240*n^4 + 836*n^3 - 1257*n^2 + 843*n - 220)*a(n-1) - (n-1)*(60*n^4 - 600*n^3 + 2094*n^2 - 3051*n + 1600)*a(n-2) + (n-2)*(n-1)*(80*n^3 - 720*n^2 + 2076*n - 1935)*a(n-3) - (n-3)*(n-2)*(n-1)*(60*n^2 - 420*n + 719)*a(n-4) + 24*(n-4)^2*(n-3)*(n-2)*(n-1)*a(n-5) + 725*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6).
a(n) ~ sqrt(3/2 + 2^(1/3) + 1/(3*2^(1/3))) * (1 + 3/2^(1/3))^n / (2*Pi*n). (End)

A349002 The number of Lyndon words of size n from an alphabet of 4 letters and 1st, 2nd and 3rd letter of the alphabet with equal frequency in the words.

Original entry on oeis.org

1, 1, 0, 2, 6, 12, 34, 120, 354, 1082, 3636, 12270, 40708, 139062, 484866, 1692268, 5944470, 21134808, 75625330, 271720146, 982116648, 3569558058, 13025614962, 47714385708, 175470892468, 647508620070, 2396613522804, 8896422981608, 33114570409896, 123566641829256

Offset: 0

R. J. Mathar, Nov 05 2021

nonn

Counts a subset of the Lyndon words in A027377. Here there is no requirement of how often the 4th letter of the alphabet occurs in the admitted word, only on the frequency of the 1st to 3rd letter of the alphabet.

			Examples for the alphabet {0,1,2,3}:
a(0)=1 counts (), the empty word.
a(3)=2 counts (021) (012).
a(4)=6 counts (0321) (0231) (0312) (0132) (0213) (0123).
a(5)=12 counts (03321) (03231) (02331) (03312) (03132) (01332) (03213) (02313) (03123) (01323) (02133) (01233).
a(6)=34 counts (020211) (002211) (012021) (002121) (010221) (001221) (033321) (033231) (032331) (023331) (012102) (011202) (002112) (010212) (001212) (033312) (011022) (010122) (001122) (033132) (031332) (013332) (033213) (032313) (023313) (033123) (031323) (013323) (032133) (023133) (031233) (013233) (021333) (012333).

Index entries for sequences related to Lyndon words

Cf. A027377, A060165, A344560.

a(n) = if(n>0, sumdiv(n, d, moebius(n/d)*sum(k=0, d\3, d!/(k!^3*(d-3*k)!)))/n, n==0) \\ Andrew Howroyd, Jan 14 2023

n*a(n) = Sum_{d|n} mu(d)*A344560(n/d) where mu = A008683.

Terms corrected and extended by Andrew Howroyd, Jan 14 2023

cp's OEIS Frontend

A361637 Constant term in the expansion of (1 + x + y + z + 1/(x*y*z))^n.

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A208425 Expansion of Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)/(1-x)^(3*n+1).

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A361703 Constant term in the expansion of (1 + w + x + y + z + 1/(w*x*y*z))^n.

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A361675 Constant term in the expansion of (1 + x*y*z + w*y*z + w*x*z + w*x*y + 1/(w*x*y*z))^n.

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A361657 Constant term in the expansion of (1 + x^2 + y^2 + 1/(x*y))^n.

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A361673 Constant term in the expansion of (1 + x*y + y*z + z*x + 1/(x*y*z))^n.

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A361677 Constant term in the expansion of (1 + x + y + z + 1/(x*y) + 1/(y*z) + 1/(z*x))^n.

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A361699 Constant term in the expansion of (1 + x^3 + y^3 + 1/(x*y))^n.

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A361637 Constant term in the expansion of (1 + x + y + z + 1/(xyz))^n.

A208425 Expansion of Sum_{n>=0} (3n)!/n!^3 x^(2n)/(1-x)^(3n+1).

A361703 Constant term in the expansion of (1 + w + x + y + z + 1/(wxy*z))^n.

A361675 Constant term in the expansion of (1 + xyz + wyz + wxz + wxy + 1/(wxy*z))^n.

A361673 Constant term in the expansion of (1 + xy + yz + zx + 1/(xy*z))^n.

A361677 Constant term in the expansion of (1 + x + y + z + 1/(xy) + 1/(yz) + 1/(z*x))^n.