A361675 -id:A361675 - cp's OEIS frontend

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

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

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)

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

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

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

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

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

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cp's OEIS Frontend

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

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

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.