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.

Previous Showing 11-15 of 15 results.

A383837 a(n) = (3*n)!/n! * [x^(3*n)] sinh(x)^n.

Original entry on oeis.org

1, 1, 16, 820, 87296, 15857205, 4390088704, 1721255653656, 907673633095680, 619593964021650475, 531571294549842067456, 559896149105493602658256, 710322778732936488128872448, 1068386732538408106621063668220, 1879866814874817967233600382304256
Offset: 0

Views

Author

Seiichi Manyama, May 11 2025

Keywords

Links

Crossrefs

Main diagonal of A381512.
Cf. A298851, A381459.

Programs

  • Mathematica
    Join[{1}, Table[Sum[(-1)^k * (n-2*k)^(3*n) * Binomial[n, k] / (2^n*n!), {k,0,n}],{n,1,20}]] (* Vaclav Kotesovec, May 13 2025 *)
  • PARI
    a(n) = sum(k=0, n, (-1)^k*(n-2*k)^(3*n)*binomial(n, k))/(2^n*n!);

Formula

a(n) = [x^n] 1/Product_{k=0..floor(n/2)} (1 - (n-2*k)^2*x).
a(n) = (1/(2^n*n!)) * Sum_{k=0..n} (-1)^k * (n-2*k)^(3*n) * binomial(n,k).
a(n) ~ c * d^n * n^(2*n - 1/2), where d = 1.35572032955623014748562257137412853926900571707993382361... and c = 0.81034327454108346293530087910356437429774959841653144433... - Vaclav Kotesovec, May 13 2025
In closed form, a(n) ~ r^(r*n) * (1 + 2*r)^(3*n+1) * exp(n) * n^(2*n - 1/2) / (sqrt(Pi*(1 - 8*r - 8*r^2)) * 2^(n - 1/2) * (1+r)^((1+r)*n)), where r = 0.002562299585216598238663221142585901101711497682846... is the positive real root of the equation exp(2*arctanh(1 + 2*r) - 6/(1 + 2*r)) = -1. - Vaclav Kotesovec, May 17 2025

A384092 a(n) = [x^n] Product_{k=1..n} 1/(1 - k^2*x)^n.

Original entry on oeis.org

1, 1, 67, 19316, 14842986, 23959995900, 70300141076691, 340026368533209120, 2526875675012579004324, 27358621384723375076245950, 414013875603209906596527455633, 8469874364125222067804767445806552, 227937197746419681734617268030982470980, 7887251806534473871432104574423885714752540
Offset: 0

Views

Author

Vaclav Kotesovec, May 19 2025

Keywords

Crossrefs

Cf. A351508, A298851, A384091.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1-k^2*x)^n, {k, 0, n}], {x, 0, n}], {n, 0, 15}]

Formula

a(n) ~ exp(n + 12/5) * n^(3*n - 1/2) / (sqrt(2*Pi) * 3^n).

A348088 a(n) = [x^n] Product_{k=1..n} 1/(1 - (2*k-1)^2 * x).

Original entry on oeis.org

1, 1, 91, 24970, 14057043, 13444400190, 19558289594910, 40250341173506100, 111335096965772406915, 398473840263173643939190, 1791905773077609090895008106, 9890754761467721759394797416396, 65747198205879568307026776928408110
Offset: 0

Views

Author

Seiichi Manyama, Sep 28 2021

Keywords

Crossrefs

Cf. A001818, A298851, A348082, A348087, A381512.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 - (2*k-1)^2*x), {k, 1, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 16 2021 *)
  • PARI
    a(n) = polcoef(1/prod(k=1, n, 1-(2*k-1)^2*x+x*O(x^n)), n);

Formula

From Vaclav Kotesovec, Oct 16 2021, updated May 16 2025: (Start)
a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 52.447924272991536496097233490380538810534457762204101802471270109895148... and c = 0.028365099209561232079163758339093959048662789595134609351298413762...
In closed form, a(n) ~ 2^(2*n) * exp(2*n) * r^(n*r + 1/2) * (1+r)^(4*n) * n^(2*n - 1/2) / (sqrt(Pi*(1 - r*(2+r))) * (2+r)^((2+r)*n - 1/2)), where r = 0.044382033760833484984013906344747760869028157215190550759633... is the root of the equation exp(4/(1+r)) = (1 + 2/r). (End)
a(n) = A381512(n,2*n-1) = (1/(2^(2*n-2)*(2*n-1)!)) * Sum_{k=0..n-1} (-1)^k * (2*n-1-2*k)^(4*n-1) * binomial(2*n-1,k) for n > 0. - Seiichi Manyama, May 16 2025

A383929 a(n) = Sum_{k=0..n} (-1)^k * binomial(2*n, k) * (n-k)^(3*n).

Original entry on oeis.org

1, 1, 60, 16626, 12640320, 20421928750, 60233972198400, 293230314199497444, 2192804991244707840000, 23869875368184417393486678, 362747302615636095725568000000, 7442995512384107947406685870219196, 200637069747857913587015560318156800000, 6945549555749361962465324588957867814958924
Offset: 0

Views

Author

Vaclav Kotesovec, May 15 2025

Keywords

Crossrefs

Cf. A002674, A298851, A383916, A383930.

Programs

  • Mathematica
    Join[{1}, Table[Sum[(-1)^(n-k)*Binomial[2*n, n-k]*k^(3*n), {k, 0, n}], {n, 1, 15}]]

Formula

a(n) ~ 2^(2*n + 1/2) * r^(3*n + 1) * n^(3*n) / (sqrt(3 - r^2) * exp(3*n) * (r^2 - 1)^n), where r = 1.1647414545521878292908344008181647954486720209245020743652... is the root of the equation (1 + r)/(1 - r) = -exp(3/r).

A383930 a(n) = Sum_{k=0..n} (-1)^k * binomial(2*n, k) * (n-k)^(5*n).

Original entry on oeis.org

1, 1, 1020, 14152314, 1071646712640, 286802348769420190, 209974096349134108992000, 355016116241074708829385321492, 1228958111984894631846657261766656000, 7960240318398277162915923478914410838135990, 89961580311571094335785117669395413813764096000000
Offset: 0

Views

Author

Vaclav Kotesovec, May 15 2025

Keywords

Comments

In general, for m>2, Sum_{k=0..n} (-1)^(n-k) * binomial(2*n, n-k) * k^(m*n) ~ 2^(2*n + 1/2) * r^(m*n + 1) * n^(m*n) / (sqrt(m + (2-m)*r^2) * exp(m*n) * (r^2 - 1)^n), where r is the root of the equation (1 + r)/(1 - r) = -exp(m/r).

Crossrefs

Cf. A002674 (m=2), A383929 (m=3), A298851*A002674 (m=4).
Cf. A383917.

Programs

  • Mathematica
    Join[{1}, Table[Sum[(-1)^(n-k)*Binomial[2*n, n-k]*k^(5*n), {k, 0, n}], {n, 1, 12}]]

Formula

a(n) ~ 2^(2*n + 1/2) * r^(5*n + 1) * n^(5*n) / (sqrt(5 - 3*r^2) * exp(5*n) * (r^2 - 1)^n), where r = 1.0145858159274292356581282820876562174881159476120290450838... is the root of the equation (1 + r)/(1 - r) = -exp(5/r).
Previous Showing 11-15 of 15 results.

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