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.

A385425 Expansion of e.g.f. exp( -LambertW(-arcsinh(x)) ).

Original entry on oeis.org

1, 1, 3, 15, 113, 1145, 14499, 220703, 3932865, 80342577, 1851286755, 47510525007, 1344106404849, 41562628517865, 1394711974335939, 50480840239135455, 1960392617938419969, 81309789407316485217, 3587373056789171999811, 167762667997938465311247
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2025

Keywords

Links

Crossrefs

Cf. A001147, A385428.
Cf. A219503, A385343, A385369, A385424.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = exp( arcsinh(x) * A(x) ).
E.g.f. A(x) satisfies A(x) = ( x + sqrt(x^2 + 1) )^A(x).
a(n) = Sum_{k=0..n} (k+1)^(k-1) * i^(n-k) * A385343(n,k), where i is the imaginary unit.
From Vaclav Kotesovec, Jun 28 2025: (Start)
a(n) ~ 2^n * exp((exp(-1) - 1)*n + 3/2) * n^(n-1) / (sqrt(1 + exp(2*exp(-1))) * (exp(2*exp(-1)) - 1)^(n - 1/2)).
Equivalently, a(n) ~ n^(n-1) / (sqrt(cosh(exp(-1))) * sinh(exp(-1))^(n - 1/2) * exp(n - 3/2)). (End)

A385428 E.g.f. A(x) satisfies A(x) = exp( arcsinh(x * A(x)) / A(x) ).

Original entry on oeis.org

1, 1, 1, 0, -11, -80, -219, 3416, 68265, 550656, -3285975, -194101248, -3177823395, -5431320960, 1202586098637, 35658624599040, 359507959906641, -12186663090266112, -677861502762897711, -13768767870225444864, 126162451289700276165, 19553934035547470168064
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2025

Keywords

Crossrefs

Cf. A001147, A385425.
Cf. A381147, A385343, A385369.

Programs

  • PARI
    a385343(n, k) = my(x='x+O('x^(n+1))); n!*polcoef(asin(x)^k/k!, n);
a(n) = sum(k=0, n, (n-k+1)^(k-1)*I^(n-k)*a385343(n, k));

Formula

E.g.f. A(x) satisfies A(x) = ( x*A(x) + sqrt((x*A(x))^2 + 1) )^(1/A(x)).
a(n) = Sum_{k=0..n} (n-k+1)^(k-1) * i^(n-k) * A385343(n,k), where i is the imaginary unit.

A385440 E.g.f. A(x) satisfies A(x) = exp( arcsinh(x * A(x)^2) ).

Original entry on oeis.org

1, 1, 5, 48, 693, 13440, 328185, 9676800, 334639305, 13284311040, 595505854125, 29756856729600, 1640160546688125, 98860780014796800, 6469121228247302625, 456736803668361216000, 34607895888408878660625, 2801319062499282124800000, 241247999301688986945463125
Offset: 0

Views

Author

Seiichi Manyama, Jun 29 2025

Keywords

Crossrefs

Cf. A001147, A385369, A385441, A385442.
Cf. A381415, A385343.

Programs

  • PARI
    a(n) = 2^n*n!*binomial((3*n+1)/2, n)/(3*n+1);

Formula

E.g.f. A(x) satisfies A(x) = (1 + 2*x*A(x)^3)^(1/2).
a(n) = 2^n * n! * binomial((3*n+1)/2,n)/(3*n+1).
a(n) = Sum_{k=0..n} (2*n+1)^(k-1) * i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) ~ 3^(3*n/2) * n^(n-1) / exp(n). - Vaclav Kotesovec, Jul 04 2025

A385441 E.g.f. A(x) satisfies A(x) = exp( arcsinh(x * A(x)^3) ).

Original entry on oeis.org

1, 1, 7, 99, 2145, 62985, 2340135, 105306075, 5568833025, 338526428625, 23261601738375, 1783052341945875, 150846228128621025, 13961656447904590425, 1403387191229030382375, 152244874971071908900875, 17729607712540283209274625, 2206069759660369525039742625, 292095560880436494680262138375
Offset: 0

Views

Author

Seiichi Manyama, Jun 29 2025

Keywords

Crossrefs

Cf. A001147, A385369, A385440, A385442.
Cf. A385343.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = (1 + 2*x*A(x)^4)^(1/2).
a(n) = 2^n * n! * binomial((4*n+1)/2,n)/(4*n+1).
a(n) = Sum_{k=0..n} (3*n+1)^(k-1) * i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) ~ 2^(3*n-1) * n^(n-1) / exp(n). - Vaclav Kotesovec, Jul 04 2025

A385442 E.g.f. A(x) satisfies A(x) = exp( arcsinh(x * A(x)^4) ).

Original entry on oeis.org

1, 1, 9, 168, 4845, 190080, 9454725, 570286080, 40454959545, 3300640358400, 304513870485825, 31348317192192000, 3562533636856719525, 443003419150516224000, 59834227558379509360125, 8722929933255903805440000, 1365222778354029313094000625, 228317457245013328565108736000
Offset: 0

Views

Author

Seiichi Manyama, Jun 29 2025

Keywords

Crossrefs

Cf. A001147, A385369, A385440, A385441.
Cf. A385343.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = (1 + 2*x*A(x)^5)^(1/2).
a(n) = 2^n * n! * binomial((5*n+1)/2,n)/(5*n+1).
a(n) = Sum_{k=0..n} (4*n+1)^(k-1) * i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) ~ 5^(5*n/2) * n^(n-1) / (exp(n) * 3^(3*n/2 + 1)). - Vaclav Kotesovec, Jul 04 2025
Showing 1-5 of 5 results.

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