A361657 -id:A361657 - cp's OEIS frontend

A361658 Constant term in the expansion of (1 + x^3 + y^3 + z^3 + 1/(xyz))^n.

1, 1, 1, 1, 1, 1, 121, 841, 3361, 10081, 25201, 55441, 194041, 1287001, 7927921, 38438401, 152312161, 516079201, 1627691521, 5745472321, 25999820401, 133086258481, 651284938921, 2860955078521, 11312609403481, 42039298455001, 158864460354601, 658342633033801

Offset: 0

Seiichi Manyama, Mar 19 2023

nonn

Cf. A002426, A361657.

Cf. A001421, A361637.

Table[n!*Sum[1/(k!^3 * (3*k)! * (n-6*k)!), {k, 0, n/6}], {n, 0, 30}] (* Vaclav Kotesovec, Mar 20 2023 *)

a(n) = n!*sum(k=0, n\6, 1/(k!^3*(3*k)!*(n-6*k)!));

a(n) = n! * Sum_{k=0..floor(n/6)} 1/(k!^3 * (3*k)! * (n-6*k)!) = Sum_{k=0..floor(n/6)} binomial(n,6*k) * A001421(k).
From Vaclav Kotesovec, Mar 20 2023: (Start)
Recurrence: (n-4)*(n-2)*n^3*a(n) = (6*n^5 - 45*n^4 + 112*n^3 - 123*n^2 + 68*n - 15)*a(n-1) - 3*(n-1)*(5*n^4 - 40*n^3 + 111*n^2 - 132*n + 59)*a(n-2) + 2*(n-2)*(n-1)*(10*n^3 - 75*n^2 + 181*n - 144)*a(n-3) - (n-3)*(n-2)*(n-1)*(15*n^2 - 90*n + 133)*a(n-4) + 3*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 7)*a(n-5) + 1727*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6).
a(n) ~ (1 + 2*sqrt(3))^(n + 3/2) / (4 * 3^(1/4) * 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)

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

cp's OEIS Frontend

A361658 Constant term in the expansion of (1 + x^3 + y^3 + z^3 + 1/(xyz))^n.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

cp's OEIS Frontend

A361658 Constant term in the expansion of (1 + x^3 + y^3 + z^3 + 1/(x*y*z))^n.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A361658 Constant term in the expansion of (1 + x^3 + y^3 + z^3 + 1/(xyz))^n.

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