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.

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A385426 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-arcsin(x)) ).

Original entry on oeis.org

1, 1, 3, 17, 145, 1665, 24115, 422305, 8681985, 205042625, 5471351875, 162811832625, 5345929731025, 192007183247425, 7488448738333875, 315170338129570625, 14238153926819850625, 687220571240324330625, 35293921478604240911875, 1921751625123502012140625
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2025

Keywords

Crossrefs

Cf. A385424, A385427.
Cf. A001147, A381145, A227464, A385343.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = exp( arcsin(x*A(x)) ).
a(n) = Sum_{k=0..n} (n+1)^(k-1) * A385343(n,k).

A385367 Expansion of e.g.f. 1/(1 - 2 * arcsinh(x)).

Original entry on oeis.org

1, 2, 8, 46, 352, 3378, 38912, 522702, 8024064, 138586722, 2659565568, 56141737518, 1292851544064, 32253357421842, 866534937329664, 24943658876605902, 765883864848531456, 24985882009464388290, 863077992845681885184, 31469256501815056673070
Offset: 0

Views

Author

Seiichi Manyama, Jun 26 2025

Keywords

Crossrefs

Cf. A296675, A385368.
Cf. A007289, A385343, A385346, A385371.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(1-2ArcSinh[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 14 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-2*asinh(x))))

Formula

E.g.f.: 1/(1 - 2 * log(x + sqrt(x^2 + 1))).
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A385371.
a(n) = Sum_{k=0..n} 2^k * k! * i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) ~ sqrt(Pi) * (1 + exp(1)) * 2^(n - 1/2) * n^(n + 1/2) / ((exp(1) - 1)^(n+1) * exp(n/2)). - Vaclav Kotesovec, Jun 27 2025

A385376 Expansion of e.g.f. 1/(1 - 2 * arcsin(x))^(1/2).

Original entry on oeis.org

1, 1, 3, 16, 117, 1104, 12687, 172320, 2698377, 47880960, 949330203, 20801387520, 499149710205, 13018307696640, 366673138800615, 11092295404707840, 358685609335654545, 12346621534211604480, 450741642786156589875, 17395372731952677519360, 707614393333663454022405
Offset: 0

Views

Author

Seiichi Manyama, Jun 27 2025

Keywords

Crossrefs

Cf. A189780, A385377.
Cf. A001147, A385343, A385346.

Programs

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

Formula

a(n) = Sum_{k=0..n} A001147(k) * A385343(n,k).
a(n) ~ sqrt(cos(1/2)) * n^n / (exp(n) * sin(1/2)^(n + 1/2)). - Vaclav Kotesovec, Jun 27 2025

A385420 Expansion of e.g.f. 1/(1 - arcsinh(3*x))^(1/3).

Original entry on oeis.org

1, 1, 4, 19, 136, 1849, 28576, 383347, 6054016, 162756433, 4512553984, 94198960723, 2151597168640, 94600222614793, 3958651982848000, 103976698299157747, 2765446240371834880, 197818347558313860385, 11750108763413970288640, 335351034570439348695955
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2025

Keywords

Comments

a(28) = -1984619795429736510626124031150165852160.

Crossrefs

Cf. A296675, A385419.
Cf. A007559, A385372, A385343, A385422.

Programs

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

Formula

E.g.f.: 1/(1 - log(3*x + sqrt(9*x^2 + 1)))^(1/3).
a(n) = Sum_{k=0..n} A007559(k) * (3*i)^(n-k) * A385343(n,k), where i is the imaginary unit.

A385422 Expansion of e.g.f. 1/(1 - arcsin(3*x))^(1/3).

Original entry on oeis.org

1, 1, 4, 37, 424, 6889, 129376, 3004597, 78196864, 2363157937, 78520720384, 2924352594373, 118146438461440, 5232528466643737, 248845526415892480, 12778931460471237397, 699044652076991610880, 40846771050451091426785, 2526020027235443981025280
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2025

Keywords

Crossrefs

Cf. A189780, A385421.
Cf. A007559, A007788, A235135, A385343, A385420.

Programs

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

Formula

a(n) = Sum_{k=0..n} A007559(k) * 3^(n-k) * A385343(n,k).
a(n) ~ sqrt(2*Pi) * cos(1)^(1/3) * 3^n * n^(n - 1/6) / (Gamma(1/3) * sin(1)^(n + 1/3) * exp(n)). - Vaclav Kotesovec, Jun 28 2025

A385427 E.g.f. A(x) satisfies A(x) = exp( arcsin(x * A(x)) / A(x) ).

Original entry on oeis.org

1, 1, 1, 2, 13, 100, 861, 9536, 127737, 1938896, 33240185, 639683552, 13601898245, 316356906944, 7998251969813, 218420230243840, 6405441641302641, 200779795515236608, 6699317212660139761, 237070134772942395904, 8868209937245857514365, 349657703494298519409664
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2025

Keywords

Crossrefs

Cf. A385424, A385426.
Cf. A381148, A385343.

Programs

  • Mathematica
    nmax = 20; A[] = 1; Do[A[x] = E^(ArcSin[x*A[x]]/A[x]) + O[x]^j // Normal, {j, 1, nmax + 1}]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 05 2025 *)
  • 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)*a385343(n, k));

Formula

a(n) = Sum_{k=0..n} (n-k+1)^(k-1) * A385343(n,k).
a(n) ~ s*(1 - r^2*s^2)^(3/4) * n^(n-1) / (sqrt(r^2*s^2*(2 + r*sqrt(1 - r^2*s^2) - r^2*s^2) - 1) * exp(n) * r^(n - 1/2)), where r = 0.4947196925654744939290429342422921705036054462455... and s = 1.929162378596122962197524561455700427559144822670... are the roots of the system of equations exp(arcsin(r*s)/s) = s, r*s/sqrt(1 - r^2*s^2) - arcsin(r*s) = s. - Vaclav Kotesovec, Jul 05 2025

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
Previous Showing 11-20 of 26 results. Next

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