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 1-6 of 6 results.

A349683 E.g.f. satisfies: log(A(x)) = exp(x*A(x)^3) - 1.

Original entry on oeis.org

1, 1, 8, 131, 3303, 113137, 4909829, 258275887, 15974450676, 1136164798581, 91366516437475, 8197719659916195, 811910298234609913, 87984131560596043801, 10355660409438349522396, 1315550191540192189444535, 179413108433279983993509731
Offset: 0

Views

Author

Seiichi Manyama, Nov 25 2021

Keywords

Crossrefs

Programs

  • Mathematica
    nterms=20;Table[Sum[(3n+1)^(k-1)*StirlingS2[n,k],{k,0,n}],{n,0,nterms-1}] (* Paolo Xausa, Nov 25 2021 *)
  • PARI
    a(n) = sum(k=0, n, (3*n+1)^(k-1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (3*n+1)^(k-1) * Stirling2(n,k).
a(n) ~ s * n^(n-1) / (3 * sqrt(1 + r*s^3) * exp(n) * r^n), where r = LambertW(1/3)/exp(1/LambertW(1/3) - 3) = 0.106691814639676411952403096776061... and s = exp(1/(3*LambertW(1/3)) - 1) = 1.341591995635184131204677967393502... are roots of the system of equations exp(r*s^3) = 1 + log(s), 3*r*s^3*exp(r*s^3) = 1. - Vaclav Kotesovec, Nov 26 2021

A349599 E.g.f. satisfies: log(A(x)) = 1 - exp(-x*A(x)^2).

Original entry on oeis.org

1, 1, 4, 29, 305, 4192, 70875, 1416781, 32551650, 841273527, 24032201213, 747395938962, 24946766300549, 880465276003861, 32274320771151308, 1197240324544640433, 42849289206116498093, 1304855947753532683776, 14954863230501575196551, -2798084168801754024136463
Offset: 0

Views

Author

Seiichi Manyama, Nov 22 2021

Keywords

Comments

a(19) < 0.

Crossrefs

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*(2*n+1)^(k-1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * (2*n+1)^(k-1) * Stirling2(n,k).

A362467 E.g.f. satisfies log(A(x)) = exp(x / A(x)^2) - 1.

Original entry on oeis.org

1, 1, -2, 11, -97, 1162, -17401, 309297, -6284804, 141430821, -3411964025, 84469913200, -1971020234987, 31982106694501, 703663251081166, -131978337454653865, 11571772746664732291, -879307513026396837470, 64266964230133042267891
Offset: 0

Views

Author

Seiichi Manyama, May 26 2023

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=0, n, (-2*n+1)^(k-1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (-2*n+1)^(k-1) * Stirling2(n,k).

A363302 E.g.f. satisfies log(A(x)) = exp(x / A(x)^3) - 1.

Original entry on oeis.org

1, 1, -4, 41, -681, 15667, -460903, 16519141, -698242716, 34004778783, -1874858325725, 115438582354977, -7851013349413919, 584508287058281419, -47281383017104676456, 4129206143361098225405, -387216724567657721607901, 38806186875022459923785751
Offset: 0

Views

Author

Seiichi Manyama, May 26 2023

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=0, n, (-3*n+1)^(k-1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (-3*n+1)^(k-1) * Stirling2(n,k).

A375868 E.g.f. satisfies A(x) = exp( 2 * (exp(x*A(x)) - 1) ).

Original entry on oeis.org

1, 2, 14, 178, 3342, 83594, 2620998, 98968034, 4375295390, 221781470202, 12684194298998, 808136496137810, 56767509202678094, 4359070656483638762, 363283064756899367462, 32658326649544884611010, 3150270056733608259143422, 324571774149991316277596378
Offset: 0

Views

Author

Seiichi Manyama, Sep 01 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = 2*sum(k=0, n, (2*n+2)^(k-1)*stirling(n, k, 2));

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A349598.
a(n) = 2 * Sum_{k=0..n} (2*n+2)^(k-1) * Stirling2(n,k).

A375869 E.g.f. satisfies A(x) = exp( 3 * (exp(x*A(x)^(2/3)) - 1) ).

Original entry on oeis.org

1, 3, 24, 327, 6405, 164856, 5276523, 202365351, 9055962270, 463552982301, 26725378964169, 1714193590625478, 121100759112660789, 9344673700445352639, 782093803535217656256, 70570503124491323693523, 6829746633650550406177713, 705729134240394228512985960
Offset: 0

Views

Author

Seiichi Manyama, Sep 01 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = 3*sum(k=0, n, (2*n+3)^(k-1)*stirling(n, k, 2));

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A349598.
a(n) = 3 * Sum_{k=0..n} (2*n+3)^(k-1) * Stirling2(n,k).
Showing 1-6 of 6 results.