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 1-8 of 8 results.

A381173 Expansion of e.g.f. (1/x) * Series_Reversion( x / (1 + x*cos(x)) ).

Original entry on oeis.org

1, 1, 2, 3, -24, -475, -5760, -52297, -155008, 8781705, 313344000, 6966991339, 102864807936, 18664712365, -71473582229504, -3387816787568865, -103478592573112320, -1899945146589964783, 18941335827815596032, 3808766537454425974739, 215681241589289359769600
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Links

Crossrefs

Cf. A381174, A381175, A381176.
Cf. A162653, A381171, A381181.
Cf. A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(n+1, k)*I^(n-k)*a185951(n, k))/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = 1/( 1 - x * cos(x * A(x)) ).
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(n+1,k) * i^(n-k) * A185951(n,k), where i is the imaginary unit.

A381376 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * cosh(x * A(x)^2) ).

Original entry on oeis.org

1, 1, 2, 9, 96, 1385, 22080, 403417, 8829184, 227956689, 6667822080, 215780258441, 7674505073664, 298885308910201, 12661212551163904, 578940699178779225, 28400662193828659200, 1488075298726340008097, 82965096417136263561216, 4904558063539270185865609
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A381171, A381300.
Cf. A185951, A381377.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n-k+1, k)/(2*n-k+1)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(2*n-k+1,k)/(2*n-k+1) * A185951(n,k).

A381181 Expansion of e.g.f. (1/x) * Series_Reversion( x / (1 + sin(x)) ).

Original entry on oeis.org

1, 1, 2, 5, 8, -79, -1584, -20539, -223616, -1855295, -1736960, 435730789, 14511117312, 338965239601, 6202042886144, 71638247035109, -714560796196864, -84697775518956799, -3650903032332091392, -115829159202293866939, -2739961030150105333760, -29414406825401517785039
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2025

Keywords

Links

Crossrefs

Cf. A201627, A381180, A381182.
Cf. A162653, A381171, A381173.
Cf. A136630, A196776.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(n+1, k)*I^(n-k)*a136630(n, k))/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = 1 + sin(x * A(x)).
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(n+1,k) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

A381449 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x * cosh(x))^2 ).

Original entry on oeis.org

1, 2, 10, 90, 1224, 22450, 517920, 14395514, 468414464, 17474840226, 735559614720, 34491849224602, 1783268816102400, 100786369113730898, 6182264844496971776, 409065938149354422330, 29043282491002728284160, 2202461172795524123296834, 177675452451923238962528256
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Links

Crossrefs

Cf. A381171, A381450.
Cf. A185951, A381447.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n+2, k)*a185951(n, k))/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = (1 + x*A(x) * cosh(x*A(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381447.
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(2*n+2,k) * A185951(n,k).

A381447 E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^2 * cosh(x*A(x)^2).

Original entry on oeis.org

1, 1, 4, 33, 432, 7745, 175680, 4818457, 155138816, 5738752161, 239890406400, 11184338164241, 575437530083328, 32387311520034913, 1979498673768132608, 130566701113312750665, 9244392468538216611840, 699309477932976288024257, 56289911059840766752456704
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Links

Crossrefs

Cf. A381171, A381448.
Cf. A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n+1, k)*a185951(n, k))/(2*n+1);

Formula

E.g.f.: ( (1/x) * Series_Reversion( x/(1 + x * cosh(x))^2 ) )^(1/2).
a(n) = (1/(2*n+1)) * Sum_{k=0..n} k! * binomial(2*n+1,k) * A185951(n,k).

A381172 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * A(x)^2 * cosh(x * A(x)) ).

Original entry on oeis.org

1, 1, 6, 75, 1416, 36065, 1160400, 45182347, 2066343552, 108594342369, 6449557524480, 427226389872491, 31230489190382592, 2497416890105693569, 216875134620623990784, 20324880119519860657515, 2044641793664946681446400, 219762483007148574205773377, 25134006030221243013604835328
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A162654, A215364, A381171.
Cf. A364984, A381175.
Cf. A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(n+2*k+1, k)/(n+2*k+1)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(n+2*k+1,k)/(n+2*k+1) * A185951(n,k).

A381448 E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^3 * cosh(x*A(x)^3).

Original entry on oeis.org

1, 1, 6, 75, 1464, 39065, 1324080, 54460987, 2635269504, 146681897553, 9233067686400, 648538095601451, 50289434320131072, 4267083467872455529, 393266542856236148736, 39121731305087283953115, 4178124995723585643970560, 476806534212831941528989217, 57905078072597558361906610176
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Links

Crossrefs

Cf. A381171, A381447.
Cf. A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(3*n+1, k)*a185951(n, k))/(3*n+1);

Formula

E.g.f.: ( (1/x) * Series_Reversion( x/(1 + x * cosh(x))^3 ) )^(1/3).
a(n) = (1/(3*n+1)) * Sum_{k=0..n} k! * binomial(3*n+1,k) * A185951(n,k).

A381450 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x * cosh(x))^3 ).

Original entry on oeis.org

1, 3, 24, 339, 7056, 195855, 6819840, 286105071, 14055420288, 791783681499, 50327779368960, 3563709848656683, 278223968271034368, 23744747385054558759, 2199369837961901789184, 219748696455778150645575, 23559108001707680103628800, 2697737574531326391439989171
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Links

Crossrefs

Cf. A381171, A381449.
Cf. A377554, A381443.
Cf. A185951, A381448.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(3*n+3, k)*a185951(n, k))/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = (1 + x*A(x) * cosh(x*A(x)))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A381448.
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(3*n+3,k) * A185951(n,k).
Showing 1-8 of 8 results.

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