cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Seiichi Manyama, Mar 19 2023

Keywords

Crossrefs

Cf. A002426, A361658.
Cf. A000897, A344560, A361637.

Programs

  • Mathematica
    Table[n!*Sum[1/(k!^2*(2*k)!*(n - 4*k)!), {k, 0, n/4}], {n, 0, 30}] (* Vaclav Kotesovec, Mar 20 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n\4, 1/(k!^2*(2*k)!*(n-4*k)!));

Formula

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 + x*y + y*z + z*x + 1/(x*y*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

Views

Author

Seiichi Manyama, Mar 20 2023

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    Table[n! * Sum[1/(k!^3 * (2*k)! * (n-5*k)!), {k,0,n/5}], {n,0,20}] (* Vaclav Kotesovec, Mar 22 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n\5, 1/(k!^3*(2*k)!*(n-5*k)!));

Formula

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)

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

Views

Author

Seiichi Manyama, Mar 21 2023

Keywords

Crossrefs

Cf. A361675, A361703, A361704.
Cf. A361657, A361658.

Programs

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

Formula

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

A361701 Constant term in the expansion of (1 + x^4 + y^4 + z^4 + 1/(x*y*z))^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 211, 1681, 7561, 25201, 69301, 166321, 360361, 990991, 5405401, 34834801, 187867681, 833709241, 3153281041, 10491944401, 31945216801, 97323704941, 345845431471, 1529597398561, 7451402805001, 35092646589001, 151591791651301
Offset: 0

Views

Author

Seiichi Manyama, Mar 21 2023

Keywords

Links

Crossrefs

Cf. A361637, A361658, A361673.
Cf. A361677.

Programs

  • Mathematica
    Table[Sum[(3*k)!/k!^3 * Binomial[7*k,3*k] * Binomial[n,7*k], {k,0,n/7}], {n,0,20}] (* Vaclav Kotesovec, Mar 22 2023 *)
  • PARI
    a(n) = sum(k=0, n\7, (3*k)!/k!^3*binomial(7*k, 3*k)*binomial(n, 7*k));

Formula

a(n) = Sum_{k=0..floor(n/7)} (3*k)!/k!^3 * binomial(7*k,3*k) * binomial(n,7*k).
From Vaclav Kotesovec, Mar 22 2023: (Start)
Recurrence: 8*n^3*(2*n - 7)*(4*n - 21)*(4*n - 7)*a(n) = 8*(224*n^6 - 2688*n^5 + 11550*n^4 - 22736*n^3 + 22666*n^2 - 11746*n + 2475)*a(n-1) - 56*(n-1)*(96*n^5 - 1200*n^4 + 5540*n^3 - 11982*n^2 + 12466*n - 5115)*a(n-2) + 224*(n-2)*(n-1)*(40*n^4 - 480*n^3 + 2065*n^2 - 3822*n + 2607)*a(n-3) - 56*(n-3)*(n-2)*(n-1)*(160*n^3 - 1680*n^2 + 5730*n - 6407)*a(n-4) + 112*(n-4)*(n-3)*(n-2)*(n-1)*(48*n^2 - 384*n + 757)*a(n-5) - 896*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 9)*a(n-6) + 823799*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-7).
a(n) ~ sqrt(c) * (1 + 7/2^(8/7))^n / (Pi^(3/2) * n^(3/2)), where c = 3.4855654710461411310762468259332410505173151761420224383969482891017005063... is the real root of the equation -559066901335151399 + 2527163634923732000*c - 5081793740448746496*c^2 + 5406293137205395456*c^3 - 3558495001867452416*c^4 + 1393309590535274496*c^5 - 303305489096114176*c^6 + 28296722014797824*c^7 = 0. (End)
Showing 1-4 of 4 results.

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