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 21-26 of 26 results.

A385368 Expansion of e.g.f. 1/(1 - 3 * arcsinh(x)).

Original entry on oeis.org

1, 3, 18, 159, 1872, 27567, 487152, 10043163, 236628864, 6272181243, 184725577728, 5984502588567, 211503539764224, 8097842686320423, 333891770433767424, 14750451600690993363, 695078159385543376896, 34800934548420464971635, 1844895428525714717343744
Offset: 0

Views

Author

Seiichi Manyama, Jun 26 2025

Keywords

Crossrefs

Cf. A296675, A385367.
Cf. A385343, A385347, A385372.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*asinh(x))))

Formula

E.g.f.: 1/(1 - 3 * log(x + sqrt(x^2 + 1))).
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A385372.
a(n) = Sum_{k=0..n} 3^k * k! * i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) ~ sqrt(Pi) * (1 + exp(2/3)) * 2^(n + 1/2) * n^(n + 1/2) / (3 * (exp(2/3) - 1)^(n+1) * exp(2*n/3)). - Vaclav Kotesovec, Jun 27 2025

A385377 Expansion of e.g.f. 1/(1 - 3 * arcsin(x))^(1/3).

Original entry on oeis.org

1, 1, 4, 29, 296, 3929, 64096, 1241437, 27834496, 709117073, 20232018944, 639064971293, 22138797783040, 834595012185193, 34013250713804800, 1490126154034917917, 69836524615835156480, 3486395656135414573985, 184703404516197170544640, 10349751400296465164293405
Offset: 0

Views

Author

Seiichi Manyama, Jun 27 2025

Keywords

Crossrefs

Cf. A189780, A385376.
Cf. A007559, A385343, A385347.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*asin(x))^(1/3)))

Formula

a(n) = Sum_{k=0..n} A007559(k) * A385343(n,k).
a(n) ~ sqrt(2*Pi) * cos(1/3)^(1/3) * n^(n - 1/6) / (Gamma(1/3) * 3^(1/3) * exp(n) * sin(1/3)^(n + 1/3)). - Vaclav Kotesovec, Jun 27 2025

A385419 Expansion of e.g.f. 1/(1 - arcsinh(2*x))^(1/2).

Original entry on oeis.org

1, 1, 3, 11, 57, 489, 5067, 50595, 573297, 9323985, 168823443, 2679252795, 45149256105, 1121782132665, 29930127386715, 629179051311315, 13329925622622945, 472248682257228705, 17395967794618282275, 434384524558247177835, 10095605146704332967705
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2025

Keywords

Comments

a(32) = -243211075187578815197768727974208613120575.

Crossrefs

Cf. A296675, A385420.
Cf. A001147, A385371, A385343.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-asinh(2*x))^(1/2)))

Formula

E.g.f.: 1/(1 - log(2*x + sqrt(4*x^2 + 1)))^(1/2).
a(n) = Sum_{k=0..n} A001147(k) * (2*i)^(n-k) * A385343(n,k), where i is the imaginary unit.

A385421 Expansion of e.g.f. 1/(1 - arcsin(2*x))^(1/2).

Original entry on oeis.org

1, 1, 3, 19, 153, 1689, 21867, 343995, 6114993, 124933425, 2820098643, 70897706595, 1939085791305, 57898697121225, 1859540697970875, 64312039377723915, 2371651908598754145, 93246340110716523105, 3882169166979871734435, 171024539858087082582195
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2025

Keywords

Crossrefs

Cf. A189780, A385422.
Cf. A001147, A001586, A385343.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-asin(2*x))^(1/2)))

Formula

a(n) = Sum_{k=0..n} A001147(k) * 2^(n-k) * A385343(n,k).
a(n) ~ sqrt(sin(2)) * 2^n * n^n / (exp(n) * sin(1)^(n+1)). - Vaclav Kotesovec, Jun 28 2025

A385443 Expansion of e.g.f. (1/x) * Series_Reversion( x/(3*x + sqrt(9*x^2+1))^(1/3) ).

Original entry on oeis.org

1, 1, 3, 7, -55, -1215, -8645, 150535, 6200145, 73698625, -1986309325, -119693799225, -1993326710375, 72724743316225, 5768642653648875, 123556356142594375, -5685256808745889375, -559310285769833973375, -14644269999088713108125, 813361265343230663434375
Offset: 0

Views

Author

Seiichi Manyama, Jun 29 2025

Keywords

Links

Crossrefs

Cf. A001147, A384241, A385444.
Cf. A385343.

Programs

  • PARI
    a(n) = 6^n*n!*binomial((4*n+1)/6, n)/(4*n+1);

Formula

E.g.f.: (1/x) * Series_Reversion( x * exp(-arcsinh(3*x)/3) ).
E.g.f.: ( (1/x) * Series_Reversion( x/(1 + 6*x)^(2/3) ) )^(1/4).
E.g.f. A(x) satisfies A(x) = exp( (1/3) * arcsinh(3*x*A(x)) ).
E.g.f. A(x) satisfies A(x) = (1 + 6*x*A(x)^4)^(1/6).
a(n) = 6^n * n! * binomial((4*n+1)/6,n)/(4*n+1).
a(n) = Sum_{k=0..n} (n+1)^(k-1) * (3*i)^(n-k) * A385343(n,k), where i is the imaginary unit.

A385444 Expansion of e.g.f. (1/x) * Series_Reversion( x/(4*x + sqrt(16*x^2+1))^(1/4) ).

Original entry on oeis.org

1, 1, 3, 0, -195, -2160, 21735, 1290240, 13253625, -758419200, -34777667925, 0, 59136015863925, 2148944878080000, -60019159896320625, -8741374232887296000, -200253365886518319375, 23678097149478739968000, 2107410008390562322321875, 0, -11628675802354427876266081875
Offset: 0

Views

Author

Seiichi Manyama, Jun 29 2025

Keywords

Links

Crossrefs

Cf. A001147, A384241, A385443.
Cf. A385343.

Programs

  • PARI
    a(n) = 8^n*n!*binomial((5*n+1)/8, n)/(5*n+1);

Formula

E.g.f.: (1/x) * Series_Reversion( x * exp(-arcsinh(4*x)/4) ).
E.g.f.: ( (1/x) * Series_Reversion( x/(1 + 8*x)^(5/8) ) )^(1/5).
E.g.f. A(x) satisfies A(x) = exp( (1/4) * arcsinh(4*x*A(x)) ).
E.g.f. A(x) satisfies A(x) = (1 + 8*x*A(x)^5)^(1/8).
a(n) = 8^n * n! * binomial((5*n+1)/8,n)/(5*n+1).
a(n) = Sum_{k=0..n} (n+1)^(k-1) * (4*i)^(n-k) * A385343(n,k), where i is the imaginary unit.
a(8*n+3) = 0 for n >= 0.
Previous Showing 21-26 of 26 results.

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

Content is available under The OEIS End-User License Agreement.