A361703 -id:A361703 - 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)

A361636 Diagonal of the rational function 1/(1 - vwx*yz (1 + 1/v + 1/w + 1/x + 1/y + 1/z)).

Original entry on oeis.org

1, 1, 1, 1, 121, 721, 2521, 6721, 128521, 1277641, 7539841, 32527441, 281835841, 3031468441, 23779315561, 139431015361, 962322302761, 9034098300361, 79726215362761, 569831799431881, 3952559737085401, 32660742079719601, 289694072383115401

Offset: 0

Seiichi Manyama, Mar 19 2023

nonn

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

Winston de Greef, Table of n, a(n) for n = 0..1064

Cf. A001850, A208425, A274783.
Cf. A008978.
Cf. A082489, A361703.

Table[Sum[(n + k)!/(k!^5*(n - 4*k)!), {k, 0, n/4}], {n, 0, 25}] (* Vaclav Kotesovec, Mar 19 2023 *)

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

a(n) = Sum_{k=0..floor(n/4)} (n+k)!/(k!^5 * (n-4*k)!).
G.f.: Sum_{k>=0} (5*k)!/k!^5 * x^(4*k)/(1-x)^(5*k+1).
Recurrence: n^4*(5*n - 19)*(5*n - 18)*(5*n - 17)*(5*n - 14)*(5*n - 13)*(5*n - 9)*a(n) = (5*n - 19)*(5*n - 18)*(5*n - 14)*(625*n^7 - 6125*n^6 + 23025*n^5 - 43195*n^4 + 45394*n^3 - 28716*n^2 + 10144*n - 1536)*a(n-1) - (5*n - 19)*(31250*n^9 - 568750*n^8 + 4441875*n^7 - 19516000*n^6 + 53172025*n^5 - 93366740*n^4 + 106140132*n^3 - 75781664*n^2 + 30987264*n - 5529600)*a(n-2) + (5*n - 4)*(31250*n^9 - 725000*n^8 + 7354375*n^7 - 42784750*n^6 + 157237100*n^5 - 378480620*n^4 + 596812963*n^3 - 594970390*n^2 + 340845072*n - 85743360)*a(n-3) + (5*n - 16)*(5*n - 9)*(5*n - 8)*(5*n - 4)*(78000*n^6 - 1450800*n^5 + 11179085*n^4 - 45672814*n^3 + 104341702*n^2 - 126378083*n + 63400710)*a(n-4) + (n-4)^4*(5*n - 14)*(5*n - 13)*(5*n - 12)*(5*n - 9)*(5*n - 8)*(5*n - 4)*a(n-5). - Vaclav Kotesovec, Mar 19 2023

A361704 Constant term in the expansion of (1 + w^2 + x^2 + y^2 + z^2 + 1/(wxy*z))^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 361, 2521, 10081, 30241, 75601, 166321, 1580041, 16833961, 114594481, 569368801, 2273150881, 7723366561, 30024671041, 193227592321, 1460787267601, 9492136169041, 50996729017081, 232560967743721, 973251617544361, 4464217099881001

Offset: 0

Seiichi Manyama, Mar 21 2023

Cf. A361675, A361703, A361705.

Table[Sum[(4*k)!/k!^4 * Binomial[6*k,4*k] * Binomial[n,6*k], {k,0,n/6}], {n,0,25}] (* Vaclav Kotesovec, Mar 25 2023 *)

a(n) = sum(k=0, n\6, (4*k)!/k!^4*binomial(6*k, 4*k)*binomial(n, 6*k));

a(n) = Sum_{k=0..floor(n/6)} (4*k)!/k!^4 * binomial(6*k,4*k) * binomial(n,6*k).
From Vaclav Kotesovec, Mar 25 2023: (Start)
Recurrence: (n-3)*n^4*a(n) = (6*n^5 - 30*n^4 + 50*n^3 - 45*n^2 + 21*n - 4)*a(n-1) - (n-1)*(15*n^4 - 90*n^3 + 195*n^2 - 195*n + 76)*a(n-2) + 5*(n-2)*(n-1)*(4*n^3 - 24*n^2 + 48*n - 33)*a(n-3) - 5*(n-3)*(n-2)*(n-1)*(3*n^2 - 15*n + 19)*a(n-4) + 6*(n-4)*(n-3)^2*(n-2)*(n-1)*a(n-5) + 11663*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6).
a(n) ~ sqrt(7 + 9/(4*2^(1/3)) + 433/(48*2^(2/3))) * (1 + 3*2^(2/3))^n / (Pi^2 * n^2). (End)

A361705 Constant term in the expansion of (1 + w^4 + x^4 + y^4 + z^4 + 1/(wxy*z))^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1681, 15121, 75601, 277201, 831601, 2162161, 5045041, 10810801, 54054001, 592191601, 5035670641, 31553973361, 157346607601, 660308770801, 2420415874801, 7951853614321, 24853781309281, 91246800876001, 497098157556001, 3346262924004001

Offset: 0

Seiichi Manyama, Mar 21 2023

Cf. A361675, A361703, A361704.

Cf. A361657, A361658.

Table[Sum[(4*k)!/k!^4 * Binomial[8*k,4*k] * Binomial[n,8*k], {k,0,n/8}], {n,0,30}] (* Vaclav Kotesovec, Mar 25 2023 *)

a(n) = sum(k=0, n\8, (4*k)!/k!^4*binomial(8*k, 4*k)*binomial(n, 8*k));

a(n) = Sum_{k=0..floor(n/8)} (4*k)!/k!^4 * binomial(8*k,4*k) * binomial(n,8*k).

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

A385606 Diagonal of the rational function 1/(1 - (v^3 + w^3 + x^3 + y^3 + z^3 + vwxyz)).

Original entry on oeis.org

1, 1, 1, 121, 721, 2521, 120121, 1262521, 7514641, 200655841, 2804296441, 23211542641, 443673670441, 7070369866561, 73192033638361, 1173608444069881, 19482750854113681, 235115468646608881, 3483568444035458401, 57574418930692099801, 769737183831483390601, 11118980118960559362001

Offset: 0

Seiichi Manyama, Jul 04 2025

nonn

Cf. A082489, A361636, A361703, A385607.

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

a(n) = Sum_{k=0..floor(n/3)} (n+2*k)!/(k!^5 * (n-3*k)!).

A385607 Diagonal of the rational function 1/(1 - (v^2 + w^2 + x^2 + y^2 + z^2 + vwxyz)).

Original entry on oeis.org

1, 1, 121, 721, 115921, 1254121, 175667521, 2723150641, 328524651841, 6553910658241, 694593264839761, 16751100559753561, 1592929589394223081, 44555491032952142881, 3872288533662063462481, 121957120480085202781681, 9836937778718128127534881, 341177468192261294809070401

Offset: 0

Seiichi Manyama, Jul 04 2025

nonn

Cf. A082489, A361636, A361703, A385606.

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

a(n) = Sum_{k=0..floor(n/2)} (n+3*k)!/(k!^5 * (n-2*k)!).

cp's OEIS Frontend

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

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A361636 Diagonal of the rational function 1/(1 - v*w*x*y*z * (1 + 1/v + 1/w + 1/x + 1/y + 1/z)).

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A361704 Constant term in the expansion of (1 + w^2 + x^2 + y^2 + z^2 + 1/(w*x*y*z))^n.

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A361705 Constant term in the expansion of (1 + w^4 + x^4 + y^4 + z^4 + 1/(w*x*y*z))^n.

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A385606 Diagonal of the rational function 1/(1 - (v^3 + w^3 + x^3 + y^3 + z^3 + v*w*x*y*z)).

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A385607 Diagonal of the rational function 1/(1 - (v^2 + w^2 + x^2 + y^2 + z^2 + v*w*x*y*z)).

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

A361636 Diagonal of the rational function 1/(1 - vwx*yz (1 + 1/v + 1/w + 1/x + 1/y + 1/z)).

A361704 Constant term in the expansion of (1 + w^2 + x^2 + y^2 + z^2 + 1/(wxy*z))^n.

A361705 Constant term in the expansion of (1 + w^4 + x^4 + y^4 + z^4 + 1/(wxy*z))^n.

A385606 Diagonal of the rational function 1/(1 - (v^3 + w^3 + x^3 + y^3 + z^3 + vwxyz)).

A385607 Diagonal of the rational function 1/(1 - (v^2 + w^2 + x^2 + y^2 + z^2 + vwxyz)).