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 1-5 of 5 results.

A364980 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x*A(x)^2).

Original entry on oeis.org

1, 1, 4, 33, 412, 6945, 147846, 3807601, 115151464, 4001162913, 157096369450, 6878742553881, 332361857826780, 17566215943990753, 1008161606338206334, 62440146891413434305, 4151012174991960338896, 294834882756167048975553
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A006153, A161633, A161635, A364981.

Programs

  • Mathematica
    Join[{1}, Table[n! * Sum[k^(n-k) * Binomial[2*n-k+1,k] / ((2*n-k+1)*(n-k)!), {k,0,n}], {n,1,20}]] (* Vaclav Kotesovec, Nov 18 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(2*n-k+1, k)/((2*n-k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(2*n-k+1,k)/( (2*n-k+1)*(n-k)! ).
a(n) ~ sqrt((1 + r*s^2)/(6 + 4*r*s^2)) * n^(n-1) / (exp(n) * r^(n + 1/2)), where r = 0.2190923703746024362724546703711998154573791458000... and s = 1.747404632046819382844696016554403302840973484745... are real roots of the system of equations 1 + exp(r*s^2)*r*s = s, 2*r*s^2*(s-1) = 1. - Vaclav Kotesovec, Nov 18 2023

A364938 E.g.f. satisfies A(x) = exp( x / (1 - x*A(x))^3 ).

Original entry on oeis.org

1, 1, 7, 73, 1141, 23821, 623341, 19650793, 725478601, 30714824377, 1467394945561, 78103975313101, 4583805610661245, 294093243091237669, 20479664124384110101, 1538423857251845781841, 124007828871708989798161, 10676865465119963987425009
Offset: 0

Views

Author

Seiichi Manyama, Aug 14 2023

Keywords

Crossrefs

Cf. A161630, A161635.
Cf. A364940, A364942, A364981.

Programs

  • Mathematica
    Join[{1}, Table[n! * Sum[(n-k+1)^(k-1) * Binomial[n+2*k-1,n-k]/k!, {k,0,n}], {n,1,20}]] (* Vaclav Kotesovec, Nov 18 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n, (n-k+1)^(k-1)*binomial(n+2*k-1, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (n-k+1)^(k-1) * binomial(n+2*k-1,n-k)/k!.
a(n) ~ sqrt(s*(1 + 2*r*s) / (4 + 3*r - 12*r*s + 12*r^2*s^2 - 4*r^3*s^3)) * n^(n-1) / (exp(n) * r^n), where r = 0.1811100305436879929789759231994897963241226689... and s = 1.893740207738561813713992833266450862854198944672... are real roots of the system of equations exp(r/(1 - r*s)^3) = s, 3*s*r^2 = (1 - r*s)^4. - Vaclav Kotesovec, Nov 18 2023

A363744 E.g.f. satisfies A(x) = exp(x * (1 + x * A(x))^2).

Original entry on oeis.org

1, 1, 5, 31, 313, 3981, 63841, 1223419, 27378737, 701091001, 20221662241, 649032795951, 22945630163017, 886151307346501, 37121193546044609, 1676607954371120611, 81222976991097364321, 4201418329450141471473, 231127287514383805458625
Offset: 0

Views

Author

Seiichi Manyama, Aug 17 2023

Keywords

Crossrefs

Cf. A125500, A365030.
Cf. A161635.

Programs

  • Mathematica
    Join[{1}, Table[n! * Sum[(n-k+1)^(k-1) * Binomial[2*k,n-k]/k!, {k,0,n}], {n,1,20}]] (* Vaclav Kotesovec, Nov 18 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n, (n-k+1)^(k-1)*binomial(2*k, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (n-k+1)^(k-1) * binomial(2*k,n-k)/k!.
a(n) ~ sqrt((1 + r*s)*(1 + 3*r*s) / (2*(1 + 2*r + 4*r^2*s + 2*r^3*s^2))) * n^(n-1) / (exp(n) * r^(n+1)), where r = 0.302307732979052080722256232095444259577495... and s = 2.910394288602135748195482733301939282588478379746... are real roots of the system of equations exp(r*(1 + r*s)^2) = s, 2*s*r^2*(1 + r*s) = 1. - Vaclav Kotesovec, Nov 18 2023

A372200 E.g.f. A(x) satisfies A(x) = exp( 2 * x / (1 - x * A(x)^(1/2))^2 ).

Original entry on oeis.org

1, 2, 12, 116, 1600, 28832, 643864, 17190392, 534707296, 19003345568, 760054943464, 33798503960168, 1654577248619728, 88437537019736816, 5125378381513865752, 320163561707158120568, 21445740148760729672896, 1533498858453023915309888
Offset: 0

Views

Author

Seiichi Manyama, Apr 21 2024

Keywords

Crossrefs

Cf. A161630, A372201.
Cf. A161635, A372160.

Programs

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

Formula

E.g.f.: A(x) = B(x)^2 where B(x) is the e.g.f. of A161635.
If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s ), then a(n) = r * n! * Sum_{k=0..n} (t*k+u*(n-k)+r)^(k-1) * binomial(n+(s-1)*k-1,n-k)/k!.

A382058 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^2), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 5, 67, 1465, 44541, 1735681, 82527439, 4632741905, 299875704697, 21989097804961, 1801520077445331, 163092373817762137, 16168084561101716725, 1741946677697976052577, 202668693570279026375671, 25324088113475137179021601, 3382305512670022948599733233, 480858973986045019386825360577
Offset: 0

Views

Author

Seiichi Manyama, Mar 13 2025

Keywords

Crossrefs

Cf. A161629, A382059.
Cf. A161635, A382032.
Cf. A001764, A377546, A382033.

Programs

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

Formula

Let F(x) be the e.g.f. of A377546. F(x) = log(A(x))/x = B(x*A(x))^2.
E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(x))^2 ) ).
a(n) = 2 * n! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(2*n+k,k)/((2*n+k) * (n-k-1)!) for n > 0.
Showing 1-5 of 5 results.

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