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 11-19 of 19 results.

A362396 E.g.f. satisfies A(x) = exp(x - x^2 * A(x)).

Original entry on oeis.org

1, 1, -1, -11, -11, 381, 2461, -21083, -449623, 221113, 99327961, 862237641, -24117649907, -612442461227, 3958786971413, 388794711373741, 2915530533136081, -239559177608638095, -6208842113295032015, 118603625804273873809, 8571701737898867135861
Offset: 0

Views

Author

Seiichi Manyama, Apr 20 2023

Keywords

Links

Crossrefs

Column k=2 of A362394.
Cf. A362430, A362431.
Cf. A125500.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(x^2*exp(x)))))

Formula

E.g.f.: exp(x - LambertW(x^2 * exp(x))) = LambertW(x^2 * exp(x))/x^2.
a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * (k+1)^(n-k-1) / (k! * (n-2*k)!).

A362690 E.g.f. satisfies A(x) = exp(x^2 + x * A(x)).

Original entry on oeis.org

1, 1, 5, 28, 245, 2816, 40537, 702976, 14270153, 332102656, 8719631981, 255020847104, 8222803663549, 289815184113664, 11085650268060929, 457386463819595776, 20248713707077863953, 957435459515190345728, 48157934732749633188565
Offset: 0

Views

Author

Seiichi Manyama, May 01 2023

Keywords

Comments

Essentially the same as A138293.

Links

Crossrefs

Cf. A349562, A362691.
Cf. A125500, A138293, A362736.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x^2-lambertw(-x*exp(x^2)))))

Formula

E.g.f.: -LambertW(-x * exp(x^2)) / x = exp( x^2 - LambertW(-x*exp(x^2)) ).
a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k+1)^(n-k-1) / (k! * (n-2*k)!).
a(n) ~ sqrt(1 + LambertW(2*exp(-2))) * 2^((n+1)/2) * n^(n-1) / (exp(n) * LambertW(2*exp(-2))^((n+1)/2)). - Vaclav Kotesovec, Nov 10 2023

A365030 E.g.f. satisfies A(x) = exp(x * (1 + x * A(x))^3).

Original entry on oeis.org

1, 1, 7, 55, 709, 11761, 243181, 6054763, 175803097, 5847578785, 219175994521, 9144024668131, 420340277237365, 21111584238219697, 1150333949592549541, 67589878866533749531, 4260172601206280708401, 286737199114729515029569
Offset: 0

Views

Author

Seiichi Manyama, Aug 17 2023

Keywords

Links

Crossrefs

Cf. A125500, A363744.
Cf. A364938.

Programs

  • Mathematica
    Array[#!*Sum[ (# - k + 1)^(k - 1)*Binomial[3 k, # - k]/k!, {k, 0, #}] &, 18, 0] (* Michael De Vlieger, Aug 18 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n, (n-k+1)^(k-1)*binomial(3*k, n-k)/k!);

Formula

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

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

A365035 E.g.f. satisfies A(x) = exp(x * (1 + x/A(x))).

Original entry on oeis.org

1, 1, 3, 1, -11, 61, 301, -6299, 7561, 903673, -9019079, -145636919, 4305630781, 7516191541, -2037845181371, 22442805921901, 944219385367441, -29922880660473359, -288352494154313999, 32071808922904896913, -273044292430852251899
Offset: 0

Views

Author

Seiichi Manyama, Aug 17 2023

Keywords

Links

Crossrefs

Cf. A125500, A143768, A363354, A363529, A365036, A365037.
Cf. A361090.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x+lambertw(x^2*exp(-x)))))

Formula

E.g.f.: exp( x + LambertW(x^2*exp(-x)) ).
a(n) = n! * Sum_{k=1..n} (-n+k+1)^(k-1) * binomial(k,n-k)/k! for n>0.

A365036 E.g.f. satisfies A(x) = exp(x * (1 + x/A(x)^2)).

Original entry on oeis.org

1, 1, 3, -5, -23, 521, -1829, -71021, 1319697, 5905297, -683965709, 8664974891, 311864420473, -13981842414695, 6694007756619, 16448800124183491, -448649039951220959, -13236887251789967071, 1210629233913421852387, -12065049302884271631269
Offset: 0

Views

Author

Seiichi Manyama, Aug 17 2023

Keywords

Links

Crossrefs

Cf. A125500, A143768, A363354, A363529, A365035, A365037.
Cf. A361091.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x+lambertw(2*x^2*exp(-2*x))/2)))

Formula

E.g.f.: exp( x + LambertW(2*x^2*exp(-2*x))/2 ).
a(n) = n! * Sum_{k=0..n} (-2*n+2*k+1)^(k-1) * binomial(k,n-k)/k!.

A365037 E.g.f. satisfies A(x) = exp(x * (1 + x/A(x)^3)).

Original entry on oeis.org

1, 1, 3, -11, -11, 1341, -14339, -168923, 8905065, -85313735, -4604578919, 197455645641, -273728455571, -267002430142187, 9427821270512373, 178475402982086701, -28273343910563670959, 713736314833387866225, 51907546734507018043057
Offset: 0

Views

Author

Seiichi Manyama, Aug 17 2023

Keywords

Links

Crossrefs

Cf. A125500, A143768, A363354, A363529, A365035, A365036.
Cf. A361092.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x+lambertw(3*x^2*exp(-3*x))/3)))

Formula

E.g.f.: exp( x + LambertW(3*x^2*exp(-3*x))/3 ).
a(n) = n! * Sum_{k=0..n} (-3*n+3*k+1)^(k-1) * binomial(k,n-k)/k!.

A372156 E.g.f. A(x) satisfies A(x) = exp( 2 * x * (1 + x * A(x)^(1/2)) ).

Original entry on oeis.org

1, 2, 8, 44, 328, 3032, 33964, 445580, 6727984, 114892784, 2192201044, 46233324788, 1068561369352, 26865052934696, 730137962157244, 21334636036296668, 667074635111434336, 22225983296836137440, 786215841115748129956, 29429693502599243538884
Offset: 0

Views

Author

Seiichi Manyama, Apr 20 2024

Keywords

Links

Crossrefs

Cf. A125500, A372157.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(2*x-2*lambertw(-x^2*exp(x)))))
  • PARI
    a(n, r=2, s=1, t=0, u=1) = r*n!*sum(k=0, n, (t*k+u*(n-k)+r)^(k-1)*binomial(s*k, n-k)/k!);

Formula

E.g.f.: A(x) = exp( 2*x - 2*LambertW(-x^2 * exp(x)) ).
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(s*k,n-k)/k!.
a(n) ~ sqrt(1 + LambertW(exp(-1/2)/2)) * n^(n-1) / (LambertW(exp(-1/2)/2)^(n+4) * 2^(n + 5/2) * exp(n)). - Vaclav Kotesovec, Aug 05 2025

A372157 E.g.f. A(x) satisfies A(x) = exp( 3 * x * (1 + x * A(x)^(1/3)) ).

Original entry on oeis.org

1, 3, 15, 99, 837, 8583, 104229, 1463031, 23362089, 418489227, 8318989089, 181823016147, 4335947796717, 112073459278095, 3122026815194205, 93267116190237807, 2974988801559127761, 100932108044810678547, 3629658464478098931897, 137928467585817206673291
Offset: 0

Views

Author

Seiichi Manyama, Apr 20 2024

Keywords

Links

Crossrefs

Cf. A125500, A372156.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(3*x-3*lambertw(-x^2*exp(x)))))
  • PARI
    a(n, r=3, s=1, t=0, u=1) = r*n!*sum(k=0, n, (t*k+u*(n-k)+r)^(k-1)*binomial(s*k, n-k)/k!);

Formula

E.g.f.: A(x) = exp( 3*x - 3*LambertW(-x^2 * exp(x)) ).
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(s*k,n-k)/k!.
Previous Showing 11-19 of 19 results.

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