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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A350972 E.g.f. = tan(x).

Original entry on oeis.org

0, 1, 0, 2, 0, 16, 0, 272, 0, 7936, 0, 353792, 0, 22368256, 0, 1903757312, 0, 209865342976, 0, 29088885112832, 0, 4951498053124096, 0, 1015423886506852352, 0, 246921480190207983616, 0, 70251601603943959887872, 0, 23119184187809597841473536, 0, 8713962757125169296170811392, 0
Offset: 0

Views

Author

N. J. A. Sloane, Mar 05 2022

Keywords

Comments

Normally these zeros would be omitted in an OEIS entry, but in view of its importance this is included as a pointer to the main entry A000182.

Examples

			tan(x) = x + (1/3)*x^3 + (2/15)*x^5 + (17/315)*x^7 + (62/2835)*x^9 + (1382/155925)*x^11 + (21844/6081075)*x^13 + (929569/638512875)*x^15 + ... = x + 2*x^3/3! + 16*x^5/5! + 272*x^7/7! + ...

Crossrefs

Cf. A000182, A000364, A122045.
See also A009006, A009764, A155585.

Programs

  • Maple
    ptan := proc(n) option remember;
    if irem(n, 2) = 0 then 0 else
    -add(`if`(k=0, 1, binomial(n, k)*ptan(n - k)), k = 0..n,2) fi end:
A350972 := n -> abs(ptan(n)): seq(A350972(n), n=0..29); # Peter Luschny, Jun 06 2022
  • Python
    from functools import cache
from math import comb as binomial
@cache
def ptan(n):
    return (0 if n % 2 == 0 else
    -sum(binomial(n,k)*ptan(n-k) if k > 0 else 1 for k in range(0,n+1,2)))
def A350972(n):
    t = ptan(n)
    return -t if t < 0 else t
print([A350972(n) for n in range(99)]) # Peter Luschny, Jun 06 2022

A201128 E.g.f. satisfies: A(x) = 1 + tan(x*A(x)).

Original entry on oeis.org

1, 1, 2, 8, 56, 536, 6336, 89984, 1502080, 28816000, 624183040, 15068741632, 401279717376, 11686228873216, 369481496674304, 12604143513731072, 461450753988067328, 18047560103055097856, 750979001470906073088, 33127742823550018125824, 1544257684260700601712640
Offset: 0

Views

Author

Paul D. Hanna, Dec 01 2011

Keywords

Comments

Radius of convergence of e.g.f. A(x) is r = 2*atan(Pi/2-1)/Pi = 0.3301951... where A(r) = Pi/2.
Radius of convergence of e.g.f. A(x) is not 2*atan(Pi/2-1)/Pi, but r = 0.39828426936099395926..., where r is the root of the equation cos(r + sqrt(r*(1-r))) = sqrt(r), and A(r) = 1 + sqrt((1-r)/r) = 2.22913364222177489... - Vaclav Kotesovec, Jan 11 2014

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 56*x^4/4! + 536*x^5/5! + ...
where A(x) = 1 + 8*x*A(x)/(Pi^2 - 4*x^2*A(x)^2) + 8*x*A(x)/(9*Pi^2 - 4*x^2*A(x)^2) + 8*x*A(x)/(25*Pi^2 - 4*x^2*A(x)^2) + 8*x*A(x)/(49*Pi^2 - 4*x^2*A(x)^2) + ...
The coefficients of x^n/n! in initial powers of G(x) = 1 + tan(x) begin:
G^1: [(1), 1, 0, 2, 0, 16, 0, 272, 0, 7936, ...];
G^2: [1,(2), 2, 4, 16, 32, 272, 544, 7936, ...];
G^3: [1, 3, (6), 12, 48, 168, 816, 4512, 23808, ...];
G^4: [1, 4, 12,(32), 120, 544, 2592, 15872, 96000, ...];
G^5: [1, 5, 20, 70, (280), 1400, 7520, 46720, 321280, ...];
G^6: [1, 6, 30, 132, 600, (3216), 19200, 125952, 925440, ...];
G^7: [1, 7, 42, 224, 1176, 6832, (44352), 312704, 2424576, ...];
G^8: [1, 8, 56, 352, 2128, 13568, 94976, (719872), 5907328, ...];
G^9: [1, 9, 72, 522, 3600, 25344, 191232, 1552752, (13518720), ...]; ...
where coefficients in parenthesis form initial terms of this sequence:
[1/1, 2/2, 6/3, 32/4, 280/5, 3216/6, 44352/7, 719872/8, 13518720/9, ...].

Links

Crossrefs

Cf. A009006.

Programs

  • Mathematica
    CoefficientList[1/x*InverseSeries[Series[x/(1 + Tan[x]), {x, 0, 21}], x],x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 11 2014 *)
Table[Sum[Binomial[n+1,2*k+1]*Sum[Binomial[j+n-2*k-1,n-2*k-1]*(j+n-2*k)!*2^(-j+2*k)*(-1)^(k+j)*StirlingS2[n,j+n-2*k]/(n+1),{j,0,2*k}],{k,0,n/2}],{n,0,20}] (* Vaclav Kotesovec after Vladimir Kruchinin, Jan 11 2014 *)
  • Maxima
    a(n):=if n=0 then 1 else sum(binomial(n+1,2*k+1)*sum(binomial(j+n-2*k-1,n-2*k-1)*(j+n-2*k)!*2^(-j+2*k)*(-1)^(k+j)*stirling2(n,j+n-2*k),j,0,2*k),k,0,(n)/2)/(n+1); /* Vladimir Kruchinin, Feb 03 2012 */
  • PARI
    {a(n)=n!*polcoeff(1/x*serreverse(x/(1+tan(x+x^2*O(x^n)))),n)}

Formula

a(n) = [x^n/n!] (1 + tan(x))^(n+1)/(n+1).
E.g.f. satisfies:
(1) A(x/(1 + tan(x))) = 1 + tan(x).
(2) A(x) = (1/x)*Series_Reversion(x/(1 + tan(x))).
(3) A(x) = 1 + Sum_{n>=1} 8*x*A(x) / ((2*n-1)^2*Pi^2 - 4*x^2*A(x)^2).
(4) A(x) = 1 + x*A(x)/(1 - x^2*A(x)^2/(3 - x^2*A(x)^2/(5 - x^2*A(x)^2/(7 - x^2*A(x)^2/(9 - ...))))), a recursive continued fraction.
a(n) = (Sum_{k=0..n/2} binomial(n+1, 2*k+1)*(Sum_{j=0..2*k} binomial(j+n-2*k-1, n-2*k-1)*(j+n-2*k)!*2^(-j+2*k)*(-1)^(k+j)*Stirling2(n, j+n-2*k)))/(n+1), n > 0, a(0)=1. - Vladimir Kruchinin, Feb 03 2012
a(n) ~ n^(n-1) * sqrt((1 + 1/sqrt(1/r-1))/(2*r))/ (exp(n) * r^n), where r = 0.39828426936099395926... is the root of the equation cos(r + sqrt(r*(1-r))) = sqrt(r). - Vaclav Kotesovec, Jan 11 2014

A009838 Expansion of e.g.f.: tanh(x)/(1+x).

Original entry on oeis.org

0, 1, -2, 4, -16, 96, -576, 3760, -30080, 278656, -2786560, 30298368, -363580416, 4748913664, -66484791296, 995368112128, -15925889794048, 270949991841792, -4877099853152256, 92635808324780032, -1852716166495600640
Offset: 0

Views

Author

R. H. Hardin

Keywords

Comments

Unsigned sequence satisfies a(n)=na(n-1)+b(n-1) where b(n) is defined in A009006. E.g.f. tanh(z)/(1-z). - Mario Catalani (mario.catalani(AT)unito.it), Feb 10 2003

Links

Programs

  • Maple
    G(x):= tanh(x)/(1+x): f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..20); # Zerinvary Lajos, Apr 03 2009
  • Mathematica
    With[{nn=20},CoefficientList[Series[Tanh[x]/(1+x),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jul 18 2013 *)

Formula

|a(n)| ~ n! * tanh(1). - Vaclav Kotesovec, Sep 22 2013

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997
Prior Mathematica program replaced and definition clarified by Harvey P. Dale, Jul 18 2013

A099932 Triangle read by rows: nonzero coefficients of polynomials 2^n*E(n,x), with E the Euler polynomials.

Original entry on oeis.org

1, 2, -1, 4, -4, 8, -12, 2, 16, -32, 16, 32, -80, 80, -16, 64, -192, 320, -192, 128, -448, 1120, -1344, 272, 256, -1024, 3584, -7168, 4352, 512, -2304, 10752, -32256, 39168, -7936, 1024, -5120, 30720, -129024, 261120, -158720, 2048, -11264
Offset: 0

Views

Author

Ralf Stephan, Nov 06 2004

Keywords

Examples

			1,
2*x - 1,
4*x^2 - 4*x,
8*x^3 - 12*x^2 + 2,
16*x^4 - 32*x^3 + 16*x,
32*x^5 - 80*x^4 + 80*x^2 - 16,
64*x^6 - 192*x^5 + 320*x^3 - 192*x

Crossrefs

With zeros: A004174/A004175.
Row sums and constant terms are (-1)^n*A009006.

A305710 Expansion of e.g.f. exp(sec(x)*exp(x) - 1).

Original entry on oeis.org

1, 1, 3, 11, 53, 297, 1959, 14499, 120409, 1097025, 10931771, 117685163, 1363889133, 16887554569, 222672557631, 3110742121059, 45912214062961, 713290136581697, 11636755988405555, 198800967493444875, 3549276499518132325, 66076184834921382313, 1280502976522048458647
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 08 2018

Keywords

Examples

			exp(sec(x)*exp(x) - 1) = 1 + x + 3*x^2/2! + 11*x^3/3! + 53*x^4/4! + 297*x^5/5! + 1959*x^6/6! + 14499*x^7/7! + ...

Crossrefs

Cf. A000364, A003701, A004211, A009006, A009212, A009248, A009268, A305708.

Programs

  • Maple
    a:=series(exp(sec(x)*exp(x)-1),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[Sec[x] Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[(2 I)^k EulerE[k, 1/2 - I/2] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]
Previous Showing 21-25 of 25 results.

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

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