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.

A381171 Expansion of e.g.f. (1/x) * Series_Reversion( x / (1 + x*cosh(x)) ).

Original entry on oeis.org

1, 1, 2, 9, 72, 725, 8640, 124117, 2117248, 41477193, 913305600, 22371549761, 604476094464, 17858943664861, 572524035586048, 19793963392789965, 734249332747960320, 29090332675789113617, 1225991945551031304192, 54765451909152748484857, 2584803582762012599910400
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. A162654, A215364, A381172.
Cf. A162653, A381173, 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)*a185951(n, k))/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = 1/( 1 - x * cosh(x * A(x)) ).
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(n+1,k) * A185951(n,k).

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.

A162654 E.g.f. satisfies: A(x) = 1 + x*cosh(x*A(x)).

Original entry on oeis.org

1, 1, 0, 3, 24, 65, 480, 8827, 72576, 657729, 13754880, 215578451, 2884992000, 62280478273, 1404449120256, 27032417472075, 640338738708480, 17729894860794497, 453894468727209984, 12438629293065953059
Offset: 0

Views

Author

Paul D. Hanna, Jul 09 2009

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^3/3! + 24*x^4/4! + 65*x^5/5! + 480*x^6/6! +...

Links

Crossrefs

Cf. A162653.

Programs

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

Formula

a(n) = n!*Sum_{k=0..n} C(n-k+1,k)/(n-k+1) * Sum_{j=0..k} C(k,j)/2^k*(2j-k)^(n-k)/(n-k)!.
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = n!*Sum_{k=0..n} m*C(n-k+m,k)/(n-k+m) * Sum_{j=0..k} C(k,j)/2^k*(2j-k)^(n-k)/(n-k)!.
a(n) ~ n^(n-1) * sqrt((2*s-1)/(s-1)) / (exp(n) * r^(n+1)), where r = 0.6184142504137720756... and s = 2.731257206829781545... are roots of the system of equations r^2*sinh(r*s) = 1, s = 1 + r*cosh(r*s). - Vaclav Kotesovec, Jul 15 2014

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.

A198865 E.g.f. satisfies: A(x) = 1 + sinh(x*A(x)^2).

Original entry on oeis.org

1, 1, 4, 31, 368, 5941, 121632, 3019563, 88140544, 2958267241, 112246484480, 4751313955543, 221980968007680, 11346405913579101, 629859586327810048, 37736053514310470371, 2426956220333852131328, 166775317658298155269585, 12195158366650225121427456
Offset: 0

Views

Author

Paul D. Hanna, Oct 30 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 31*x^3/3! + 368*x^4/4! + 5941*x^5/5! +...
Related expansions.
A(x)^2 = 1 + 2*x + 10*x^2/2! + 86*x^3/3! + 1080*x^4/4! + 18042*x^5/5! +...
1/(1 + sinh(x))^2 = 1 - 2*x + 6*x^2/2! - 26*x^3/3! + 144*x^4/4! - 962*x^5/5! +...
Coefficients of [x^n/n!] in the odd powers of (1 + sinh(x)) begin:
1: [(1), 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0,...];
3: [1,(3), 6, 9, 24, 63, 96, 549, 384, 4923, 1536, 44289,...];
5: [1, 5,(20), 65, 200, 725, 2720, 9665, 41600, 165125,...];
7: [1, 7, 42,(217), 1008, 4627, 22512, 112357, 567168,...];
9: [1, 9, 72, 513,(3312), 20169, 122112, 756513, 4770432,...];
11:[1, 11, 110, 1001, 8360,(65351), 492800, 3693701, 27948800,...];
13:[1, 13, 156, 1729, 17784, 171613,(1581216), 14210209,...];
15:[1, 15, 210, 2745, 33600, 387675, 4262160,(45293445),...];
17:[1, 17, 272, 4097, 58208, 783377, 10057472, 124378817,(1498389248), ...]; ...
where the coefficients in parenthesis generate this sequence like so:
[1, 3/3, 20/5, 217/7, 3312/9, 65351/11, 1581216/13, 45293445/15,...].

Crossrefs

Cf. A162653.

Programs

  • Mathematica
    CoefficientList[Sqrt[InverseSeries[Series[x/(1 + Sinh[x])^2, {x, 0, 21}], x]/x],x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 11 2014 *)
  • PARI
    /* PARI programs for a(n,m) where A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! */
{a(n,m=1)=n!*polcoeff((1+sinh(x +x*O(x^n)))^(2*n+m)*m/(2*n+m),n)}
  • PARI
    {a(n,m=1)=sum(k=0, n, m*binomial(2*n+m, k)/(2*n+m)*sum(j=0, k, (-1)^(k-j)*binomial(k, j)*(2*j-k)^n/2^k))}
  • PARI
    {a(n,m=1)=n!*polcoeff((serreverse(x/(1+sinh(x +x*O(x^n)))^2)/x)^(m/2),n)}

Formula

(1) E.g.f. satisfies: A( x/(1 + sinh(x))^2 ) = 1 + sinh(x).
(2) E.g.f.: A(x) = sqrt( Series_Reversion( x/(1 + sinh(x))^2 ) / x ).
(3) a(n) = [x^n/n!] (1 + sinh(x))^(2*n+1) / (2*n+1).
(4) a(n) = Sum_{k=0..n} C(2*n+1,k)/(2*n+1)*Sum_{j=0..k} (-1)^(k-j)*C(k,j)*(2j-k)^n/2^k.
(5) Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} m*C(2*n+m,k)/(2*n+m)*Sum_{j=0..k} (-1)^(k-j)*C(k,j)*(2j-k)^n/2^k.
a(n) ~ s*sqrt((2-2*s+s^2)/(2*(2-3*s+2*s^2))) * n^(n-1) * (2*s*sqrt(2-2*s+s^2))^n / exp(n), where s = 1.75931315231552523... is the root of the equation 2*sqrt(2+(s-2)*s) * log(1+sqrt(1+(1-s)^2)-s) = -s. - Vaclav Kotesovec, Jan 11 2014

A381177 E.g.f. A(x) satisfies A(x) = 1/( 1 - A(x) * sinh(x * A(x)) ).

Original entry on oeis.org

1, 1, 6, 73, 1352, 33861, 1072000, 41083477, 1849680768, 95708731945, 5597075177984, 365091888890433, 26281788308598784, 2069729710424907181, 177006820644852031488, 16337090667286093559821, 1618592591411194127089664, 171337824188415839421148881, 19299478529228162963028508672
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2025

Keywords

Crossrefs

Cf. A162653, A201628, A381179.
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+2*k+1, k)/(n+2*k+1)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(n+2*k+1,k)/(n+2*k+1) * A136630(n,k).

A381179 E.g.f. A(x) satisfies A(x) = 1 + sinh(x*A(x)) / A(x).

Original entry on oeis.org

1, 1, 0, 1, 8, 21, 64, 1093, 8448, 47785, 654848, 9402537, 94222336, 1264390141, 23392960512, 363389219053, 5722054885376, 117602664867921, 2434091053613056, 47867013812467921, 1080303165427679232, 26716998341391367141, 645003218568158904320, 16403742152044108508181
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2025

Keywords

Crossrefs

Cf. A162653, A201628, A381177.
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-k+1, k)/(n-k+1)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(n-k+1,k)/(n-k+1) * A136630(n,k).

A381430 E.g.f. A(x) satisfies A(x) = 1 + sinh(x*A(x)^3).

Original entry on oeis.org

1, 1, 6, 73, 1368, 34861, 1126368, 44135701, 2034072960, 107823563641, 6463383851520, 432331180935457, 31924171503581184, 2579483385868484005, 226383845487041421312, 21445302563389991287981, 2180974075392495296544768, 237009522316557393020262001, 27409082977094100068471537664
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Links

Crossrefs

Cf. A162653, A198865.
Cf. A136630, A381443.

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(3*n+1, k)*a136630(n, k))/(3*n+1);

Formula

E.g.f.: ( (1/x) * Series_Reversion( x/(1 + sinh(x))^3 ) )^(1/3).
a(n) = (1/(3*n+1)) * Sum_{k=0..n} k! * binomial(3*n+1,k) * A136630(n,k).
Showing 1-8 of 8 results.

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