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 1-8 of 8 results.

A377832 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(-x) ).

Original entry on oeis.org

1, 3, 29, 508, 13137, 452616, 19549021, 1016932512, 61940154177, 4325943203200, 340900244374461, 29927648769380352, 2896829645184711121, 306522175683831195648, 35201889560564096132925, 4360880891670519541927936, 579686447990401730151243009, 82304944815106131595482267648
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(3*n-k+1, n-k)/k!);

Formula

E.g.f. A(x) satisfies A(x) = exp(x * A(x))/(1 - x*A(x))^2.
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(3*n-k+1,n-k)/k!.
a(n) ~ (1 + sqrt(3))^(4*n + 5/2) * n^(n-1) / (3^(1/4) * 2^(3*n + 5/2) * exp((sqrt(3) - 1)*n - 2 + sqrt(3))). - Vaclav Kotesovec, Nov 09 2024

A377743 E.g.f. satisfies A(x) = exp(x) / (1 - x * A(x))^3.

Original entry on oeis.org

1, 4, 43, 853, 25141, 989581, 48885187, 2910389875, 202958554057, 16233163690537, 1465257396236551, 147359765665925143, 16341437664329027389, 1981169884084699982701, 260701144663332062732491, 37007345616327485166160651, 5637148375602304430334748945, 917186940500490837457393476817
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(n-k-1)*binomial(4*k+2, k)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k-1) * binomial(4*k+2,k)/(n-k)!.

A377810 E.g.f. satisfies A(x) = exp(x * A(x)) / (1 - x)^2.

Original entry on oeis.org

1, 3, 17, 154, 1993, 34066, 728209, 18733926, 564117425, 19473863986, 758421401401, 32901791851006, 1573602042306265, 82267318018246986, 4667656830688700801, 285662368622361581206, 18758565855176593500385, 1315663025587514658845026, 98160436697525045768511721
Offset: 0

Views

Author

Seiichi Manyama, Nov 08 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x)^2))/(1-x)^2))
    
  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(n+k+1, n-k)/k!);

Formula

E.g.f.: exp( -LambertW(-x/(1-x)^2) )/(1-x)^2.
E.g.f.: -LambertW(-x/(1-x)^2)/x.
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n+k+1,n-k)/k!.
a(n) ~ 2^(n + 3/2) * sqrt(1 + 4*exp(-1) - sqrt(1 + 4*exp(-1))) * n^(n-1) / ((-1 + sqrt(1 + 4*exp(-1)))^(3/2) * (1 + 2*exp(-1) - sqrt(1 + 4*exp(-1)))^(n + 1/2) * exp(2*n+1)). - Vaclav Kotesovec, Nov 11 2024

A377745 E.g.f. satisfies A(x) = exp(x) / (1 - x * A(x)^2)^2.

Original entry on oeis.org

1, 3, 35, 865, 32917, 1699311, 111033607, 8788108477, 817439352233, 87406186549339, 10564550856634411, 1424421297360350169, 211968687043802337469, 34509326697582566247367, 6101526326400539736369935, 1164298084658023787974823221, 238495519792465232104337607505
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (2*k+1)^(n-k-1)*binomial(5*k+1, k)/(n-k)!);

Formula

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

A377744 E.g.f. satisfies A(x) = exp(x) / (1 - x * A(x))^4.

Original entry on oeis.org

1, 5, 69, 1741, 65025, 3238401, 202252549, 15216086789, 1340493558497, 135418524663457, 15436319894361141, 1960277599669850517, 274474966233168968353, 42012725272366653895169, 6979546631782182590117189, 1250777360824265136694022341, 240516661686854988775792192833
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(n-k-1)*binomial(5*k+3, k)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k-1) * binomial(5*k+3,k)/(n-k)!.

A380764 E.g.f. A(x) satisfies A(x) = exp(x * (1 - x*A(x))) / (1 - x*A(x))^2.

Original entry on oeis.org

1, 3, 21, 283, 5825, 161281, 5616415, 235957275, 11619036385, 656499970657, 41874164431631, 2976512157543739, 233338979438666161, 20000563338051696609, 1860931002481238778511, 186799953169800497128891, 20122315691834196706830017, 2315417027513322899728489537
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n, q=1, r=1, s=0, t=-1, u=2) = q*n!*sum(k=0, n, (r*n+(s-r)*k+q)^(k-1)*binomial((r*u+1)*n+((s-r)*u+t-1)*k+q*u-1, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (n-k+1)^(k-1) * binomial(3*n-4*k+1,n-k)/k!.

A380765 E.g.f. A(x) satisfies A(x) = exp(x * (1 - x*A(x))^2) / (1 - x*A(x))^2.

Original entry on oeis.org

1, 3, 19, 241, 4853, 131601, 4466875, 182546421, 8739580841, 480023587297, 29759608788551, 2055884656223949, 156623317577663293, 13045653418406432721, 1179479817324874518419, 115042876530398843323621, 12041278143223263581774417, 1346252625757920938545507521
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n, q=1, r=1, s=0, t=-2, u=2) = q*n!*sum(k=0, n, (r*n+(s-r)*k+q)^(k-1)*binomial((r*u+1)*n+((s-r)*u+t-1)*k+q*u-1, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (n-k+1)^(k-1) * binomial(3*n-5*k+1,n-k)/k!.

A380720 E.g.f. A(x) satisfies A(x) = exp(x * A(x)^2 * (1 - x*A(x))^4) / (1 - x*A(x))^2.

Original entry on oeis.org

1, 3, 27, 427, 9829, 299421, 11399767, 522120299, 27993612745, 1721382881401, 119487832998811, 9244561661068647, 788985451618181869, 73644131873399817653, 7463589265871298367711, 816231439143125763495811, 95811879190166378655829393, 12015708296507465444922873585
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (n+k+1)^(k-1)*binomial(3*n-3*k+1, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (n+k+1)^(k-1) * binomial(3*n-3*k+1,n-k)/k!.
Showing 1-8 of 8 results.