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.

A365287 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x^3*A(x)^3).

Original entry on oeis.org

1, 1, 2, 6, 48, 720, 11520, 183960, 3185280, 65681280, 1637193600, 46436544000, 1423113753600, 46607434473600, 1648149184281600, 63369409495392000, 2634451417524326400, 117088187211284889600, 5518546983426135859200, 275022667579200532992000
Offset: 0

Views

Author

Seiichi Manyama, Aug 31 2023

Keywords

Crossrefs

Cf. A358065, A365285, A365286.
Cf. A161633, A365283.

Programs

  • Mathematica
    Join[{1},Table[n!/(n+1) * Sum[(n-3*k)^k * Binomial[n+1,n-3*k]/k!, {k,0,Floor[n/3]}], {n,1,20}]] (* Vaclav Kotesovec, Nov 08 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n\3, (n-3*k)^k*binomial(n+1, n-3*k)/k!)/(n+1);

Formula

a(n) = (n!/(n+1)) * Sum_{k=0..floor(n/3)} (n-3*k)^k * binomial(n+1,n-3*k)/k!.
a(n) ~ 3^(n/3) * (1 + 4*LambertW(3^(1/4)/4))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(3^(1/4)/4)) * 2^(8*n/3 + 4) * exp(n) * LambertW(3^(1/4)/4)^(4*n/3 + 3/2)). - Vaclav Kotesovec, Nov 08 2023

A365282 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x^2*A(x)).

Original entry on oeis.org

1, 1, 2, 12, 96, 900, 10800, 157080, 2634240, 50455440, 1089849600, 26157479040, 690848040960, 19924295751360, 623024501299200, 20996216063222400, 758724126031872000, 29267547577396128000, 1200407895406514995200
Offset: 0

Views

Author

Seiichi Manyama, Aug 31 2023

Keywords

Crossrefs

Cf. A358064, A365283, A365284.
Cf. A365285.

Programs

  • Mathematica
    Join[{1}, Table[n! * Sum[(n-2*k)^k * Binomial[n-k+1,n-2*k] / ((n-k+1)*k!), {k,0,Floor[n/2]}], {n,1,20}]] (* Vaclav Kotesovec, Aug 31 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n\2, (n-2*k)^k*binomial(n-k+1, n-2*k)/((n-k+1)*k!));

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)^k * binomial(n-k+1,n-2*k)/( (n-k+1)*k! ).
a(n) ~ sqrt((s+1)/(2*s-1)) * (s-1)^((n+1)/2) * s^(n/2 + 1) * n^(n-1) / exp(n), where s = 3.011547791499065828694160466323712196300874261862... is the root of the equation (s-1)*LambertW(2*(s-1)^2/s) = 2. - Vaclav Kotesovec, Aug 31 2023

A371018 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x^2*exp(x)) ).

Original entry on oeis.org

1, 0, 2, 6, 60, 620, 7950, 129402, 2365496, 50512968, 1208642490, 32223422990, 947694971652, 30435132773916, 1061061668979494, 39889366397571810, 1608910488000292080, 69305890226183224592, 3175519952912430375666, 154216789672147809137046
Offset: 0

Views

Author

Seiichi Manyama, Mar 08 2024

Keywords

Links

Crossrefs

Cf. A161633, A371019.
Cf. A365283.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1+x^2*exp(x)))/x))
  • PARI
    a(n) = n!*sum(k=0, n\2, k^(n-2*k)*binomial(n+1, k)/(n-2*k)!)/(n+1);

Formula

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

A365284 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x^2*A(x)^3).

Original entry on oeis.org

1, 1, 2, 12, 144, 1980, 31680, 630840, 15093120, 411883920, 12607660800, 430740858240, 16265744732160, 671629503504960, 30093198326231040, 1454898560062147200, 75503612563771392000, 4186035286381024876800, 246916968958719605145600
Offset: 0

Views

Author

Seiichi Manyama, Aug 31 2023

Keywords

Crossrefs

Cf. A358064, A365282, A365283.

Programs

  • Mathematica
    Join[{1}, Table[n! * Sum[(n - 2*k)^k*Binomial[n + k + 1, n - 2*k]/((n + k + 1)*k!), {k, 0, Floor[n/2]}], {n, 1, 20}]] (* Vaclav Kotesovec, Mar 10 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\2, (n-2*k)^k*binomial(n+k+1, n-2*k)/((n+k+1)*k!));

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)^k * binomial(n+k+1,n-2*k)/( (n+k+1)*k! ).
a(n) ~ sqrt((1 + 2*r^2*s^3) / (12*r^2*s + 9*r^4*s^4)) * n^(n-1) / (exp(n) * r^n), where s = 1.766482823850997284176450269002863328615073785089684545740773169... is the root of the equation 3*(s-1)*LambertW(2*s*(s-1)^2) = 2 and r = 1/sqrt(3*s^3*(s-1)) = 0.280882078734447087396397749882018030987007964077248... - Vaclav Kotesovec, Mar 10 2024

A370889 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*exp(x^2/2)) ).

Original entry on oeis.org

1, 1, 2, 9, 72, 735, 9000, 133035, 2325120, 46631025, 1053108000, 26484495345, 734652737280, 22280390827695, 733335188826240, 26035824337798275, 991872319953715200, 40360728513989909025, 1747119524427614937600, 80166580022376802179225
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2024

Keywords

Links

Crossrefs

Cf. A161633, A370926.
Cf. A365283.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1+x*exp(x^2/2)))/x))
  • PARI
    a(n) = n!*sum(k=0, n\2, (n-2*k)^k*binomial(n+1, n-2*k)/(2^k*k!))/(n+1);

Formula

a(n) = (n!/(n+1)) * Sum_{k=0..floor(n/2)} (n-2*k)^k * binomial(n+1,n-2*k)/(2^k * k!).
a(n) ~ (1 + 3*LambertW(1/3))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(1/3)) * 3^(3*n/2 + 2) * exp(n) * LambertW(1/3)^(3*(n+1)/2)). - Vaclav Kotesovec, Mar 06 2024

A371067 E.g.f. satisfies A(x) = 1 + x*exp(x^2*A(x)^2).

Original entry on oeis.org

1, 1, 0, 6, 48, 180, 2880, 46200, 483840, 9087120, 203212800, 3752511840, 89413632000, 2510276408640, 66301996400640, 1982685238934400, 67064515854336000, 2274167610024710400, 82881756045036748800, 3301346557970183923200, 135363022243685203968000
Offset: 0

Views

Author

Seiichi Manyama, Mar 09 2024

Keywords

Crossrefs

Cf. A365283, A370927.
Cf. A161631, A371068.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)^k * binomial(2*k+1,n-2*k)/( (2*k+1)*k! ).
a(n) ~ n^(n-1) / (sqrt(2) * exp(n) * r^(n+1)), where r = 0.450347181930267755599214125867779338412791581819135528888185619948594... and s = 2.1478259175343697310213089706837271102656629945040966643073615920885... are roots of the system of equations exp(r^2*s^2)*r = s-1, 2*(s-1)*r^2*s = 1. - Vaclav Kotesovec, Mar 10 2024
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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