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 1-3 of 3 results.

A318000 E.g.f.: log( 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))) ).

Original entry on oeis.org

1, 4, 24, 256, 3840, 73024, 1688064, 45991936, 1443102720, 51249316864, 2032187080704, 89000317321216, 4266655914393600, 222232483747938304, 12496860570760249344, 754582425618372100096, 48694058763984285204480, 3344368871374116303929344, 243577066332044464943529984, 18751361596512920229250072576
Offset: 1

Views

Author

Paul D. Hanna, Aug 20 2018

Keywords

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 24*x^3/3! + 256*x^4/4! + 3840*x^5/5! + 73024*x^6/6! + 1688064*x^7/7! + 45991936*x^8/8! + 1443102720*x^9/9! + 51249316864*x^10/10! + ...
such that cosh(x + A(x)) + sinh(x - A(x)) = 1.
RELATED SERIES.
(1) exp(A(x)) = 1 + x + 5*x^2/2! + 37*x^3/3! + 425*x^4/4! + 6601*x^5/5! + 129005*x^6/6! + 3044077*x^7/7! + 84239825*x^8/8! + ... + A318002(n)*x^n/n! + ...
which equals 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))).
(2) Let F(F(x)) = A(x) then
F(x) = x + 2*x^2/2! + 6*x^3/3! + 56*x^4/4! + 600*x^5/5! + 8432*x^6/6! + 144816*x^7/7! + 2892416*x^8/8! + 66721920*x^9/9! + ... + A318001(n)*x^n/n! + ...
where cosh(F(x) - F(-x)) - sinh(F(x) + F(-x)) = 1.

Links

Crossrefs

Cf. A318001 (A(A(x))), A318002 (exp(A(x))), A318005 (variant).

Programs

  • PARI
    {a(n) = my(A = log( 2*cosh(x +x^2*O(x^n)) / (1 + sqrt(1 - 2*sinh(2*x +x^2*O(x^n)))) )); n!*polcoeff(A,n)}
for(n=1,25,print1(a(n),", "))

Formula

E.g.f. A(x) satisfies:
(1) A(-A(-x)) = x.
(2) 1 = Sum_{n>=0} ( x + (-1)^n*A(x) )^n/n!.
(3a) 1 = cosh(A(x) + x) - sinh(A(x) - x).
(3b) 1 = cosh(x)*exp(-A(x)) + sinh(x)*exp(A(x)).
(3c) 1 = exp(x)*cosh(A(x)) - exp(-x)*sinh(A(x)).
(4a) A(x) = log( 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))) ).
(4b) A(x) = log( (1 - sqrt(1 - 2*sinh(2*x))) / (2*sinh(x)) ).
(5) A(x) = F(F(x)) where F(x) is the e.g.f. of A318001, which satisfies: 1 = cosh(F(x) - F(-x)) - sinh(F(x) + F(-x)).
a(n) ~ 5^(1/4) * 2^(n - 1/2) * n^(n-1) / (exp(n) * log((1 + sqrt(5))/2)^(n - 1/2)). - Vaclav Kotesovec, Aug 21 2018

A318006 E.g.f. A(x) satisfies: cos(A(x)) + sin(A(x)) = 1/( cos(A(-x)) + sin(A(-x)) ).

Original entry on oeis.org

1, 2, 6, 40, 360, 4592, 70896, 1279360, 26497920, 619457792, 16166151936, 466022394880, 14708199367680, 504453778491392, 18681868054910976, 742996971891097600, 31583887537425776640, 1429076863804079931392, 68575244394079858262016, 3478457209493103235563520, 185971933231431479545036800
Offset: 1

Views

Author

Paul D. Hanna, Aug 27 2018

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 40*x^4/4! + 360*x^5/5! + 4592*x^6/6! + 70896*x^7/7! + 1279360*x^8/8! + 26497920*x^9/9! + 619457792*x^10/10! + ...
such that
cos(A(x)) + sin(A(x)) = 1/( cos(A(-x)) + sin(A(-x)) )
and
sin(2*A(x)) = 2*sin(2*x)/(2 - sin(2*x)) = 2*sin(A(x))*cos(A(x)).
RELATED SERIES.
(a) A(A(x)) = x + 4*x^2/2! + 24*x^3/3! + 224*x^4/4! + 2880*x^5/5! + 48064*x^6/6! + 989184*x^7/7! + 24218624*x^8/8! + ... + A318005(n)*x^n/n! + ...
where A(A(x)) = arcsin( 2*sin(2*x)/(2 - sin(2*x)) ) /2.
(b) cos(A(x)) + sin(A(x)) = 1/(cos(A(-x)) - sin(A(-x))) = 1 + x + x^2/2! - x^3/3! - 7*x^4/4! - 59*x^5/5! - 239*x^6/6! - 421*x^7/7! + 4913*x^8/8! + 108361*x^9/9! + 1000321*x^10/10! + ...
where cos(A(x)) + sin(A(x)) = sqrt( (2 + sin(2*x))/(2 - sin(2*x)) ).

Links

Crossrefs

Cf. A318005, A318001, A200560.

Programs

  • Mathematica
    nmax = 25; Rest[CoefficientList[Series[ArcSin[4/(2 - Sin[2*x]) - 2]/2, {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Aug 28 2018 *)
  • PARI
    /* A(x) = arcsin( 2*sin(2*x)/(2 - sin(2*x)) )/2 */
{a(n) = my(A = asin( 2*sin(2*x +x*O(x^n))/(2 - sin(2*x +x*O(x^n))) )/2 ); n!*polcoeff(A,n)}
for(n=1,25, print1(a(n),", "))
  • PARI
    /* From 1 = cos(A(A(x)) + x) + sin(A(A(x)) - x) */
{a(n) = my(A=[1,1]); for(i=1, n, A = concat(A,0);
A[#A] = -Vec(cos(subst(x*Ser(A),x,x*Ser(A)) + x) + sin(subst(x*Ser(A),x,x*Ser(A)) - x))[#A+1]/2; ); n!*A[n]}
for(n=1, 25, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies:
(1) A(-A(-x)) = x.
(2a) 1 = Sum_{n>=0} (-1)^floor(n/2) * ( A(x) - (-1)^n*A(-x) )^n/n!.
(2b) 1 = Sum_{n>=0} (-1)^floor(n/2) * ( A(A(x)) + (-1)^n*x )^n/n!.
(3a) 1 = cos( A(x) - A(-x) ) + sin( A(x) + A(-x) ).
(3b) 1 = ( cos(A(x)) + sin(A(x)) ) * ( cos(A(-x)) + sin(A(-x)) ).
(4a) 1 = cos(A(A(x)) + x) + sin(A(A(x)) - x).
(4b) 1 = ( cos(A(A(x))) + sin(A(A(x))) ) * (cos(x) - sin(x)).
(5a) A(x) = arcsin( 2*sin(2*x)/(2 - sin(2*x)) )/2.
(5b) A(A(x)) = arcsin( sin(2*x)/(1 - sin(2*x)) )/2, which is the e.g.f. of A318005.
(6) cos(A(x)) + sin(A(x)) = sqrt( (2 + sin(2*x))/(2 - sin(2*x)) ).
a(n) ~ sqrt(3) * 5^(1/4) * 2^(n-2) * n^(n-1) / (exp(n) * (arcsin(2/3))^(n - 1/2)). - Vaclav Kotesovec, Aug 28 2018

A319145 E.g.f. A = A(x,m) satisfies: cn(A + x, m) + sn(A - x, m) = 1, where sn(x,m) and cn(x,m) are Jacobi elliptic functions with parameter m, as an irregular triangle of coefficients read by rows.

Original entry on oeis.org

1, 4, 24, 0, 224, -64, 2880, -1920, 0, 48064, -49984, 1024, 989184, -1365504, 129024, 0, 24218624, -40854528, 8583168, -16384, 687083520, -1352540160, 471859200, -8355840, 0, 22151148544, -49507063808, 24589796352, -1331806208, 262144, 799546834944, -1993321955328, 1286051069952, -141582532608, 536346624, 0, 31934834253824, -87721489006592, 69349000355840, -12549922078720, 198078103552, -4194304
Offset: 1

Views

Author

Paul D. Hanna, Sep 11 2018

Keywords

Examples

			E.g.f.: A(x,m) = x + 4*x^2/2! + 24*x^3/3! + (-64*m + 224)*x^4/4! + (-1920*m + 2880)*x^5/5! + (1024*m^2 - 49984*m + 48064)*x^6/6! + (129024*m^2 - 1365504*m + 989184)*x^7/7! + (-16384*m^3 + 8583168*m^2 - 40854528*m + 24218624)*x^8/8! + (-8355840*m^3 + 471859200*m^2 - 1352540160*m + 687083520)*x^9/9! + (262144*m^4 - 1331806208*m^3 + 24589796352*m^2 - 49507063808*m + 22151148544)*x^10/10! + (536346624*m^4 - 141582532608*m^3 + 1286051069952*m^2 - 1993321955328*m + 799546834944)*x^11/11! + (-4194304*m^5 + 198078103552*m^4 - 12549922078720*m^3 + 69349000355840*m^2 - 87721489006592*m + 31934834253824)*x^12/12! + ...
such that cn(A + x, m)  +  sn(A - x, m)  =  1.
This triangle of coefficients of x^n*m^k/n! in A(x,m) begins
1;
4;
24, 0;
224, -64;
2880, -1920, 0;
48064, -49984, 1024;
989184, -1365504, 129024, 0;
24218624, -40854528, 8583168, -16384;
687083520, -1352540160, 471859200, -8355840, 0;
22151148544, -49507063808, 24589796352, -1331806208, 262144;
799546834944, -1993321955328, 1286051069952, -141582532608, 536346624, 0;
31934834253824, -87721489006592, 69349000355840, -12549922078720, 198078103552, -4194304; ...
RELATED SERIES.
cn(A(x,m) + x, m) = 1 - 4*x^2/2! - 24*x^3/3! + (64*m - 224)*x^4/4! + (1920*m - 2880)*x^5/5! + (-1024*m^2 + 50944*m - 47104)*x^6/6! + (-129024*m^2 + 1405824*m - 948864)*x^7/7! + (16384*m^3 - 8798208*m^2 + 42037248*m - 22820864)*x^8/8! + (8355840*m^3 - 491212800*m^2 + 1381570560*m - 638699520)*x^9/9! + (-262144*m^4 + 1367932928*m^3 - 25781010432*m^2 + 50035417088*m - 20383842304)*x^10 + ...
sn(A(x,m) - x, m) = 1 - cn(A(x,m) + x, m) = 4*x^2/2! + 24*x^3/3! + (-64*m + 224)*x^4/4! + (-1920*m + 2880)*x^5/5! + (1024*m^2 - 50944*m + 47104)*x^6/6! + (129024*m^2 - 1405824*m + 948864)*x^7/7! + ...
Related Jacobi elliptic functions with parameter m begin:
sn(x,m) = x + (-m - 1)*x^3/3! + (m^2 + 14*m + 1)*x^5/5! + (-m^3 - 135*m^2 - 135*m - 1)*x^7/7! + (m^4 + 1228*m^3 + 5478*m^2 + 1228*m + 1)*x^9/9! + (-m^5 - 11069*m^4 - 165826*m^3 - 165826*m^2 - 11069*m - 1)*x^11/11! + ...
cn(x,m) = 1 - x^2/2! + (4*m + 1)*x^4/4! + (-16*m^2 - 44*m - 1)*x^6/6! + (64*m^3 + 912*m^2 + 408*m + 1)*x^8/8! + (-256*m^4 - 15808*m^3 - 30768*m^2 - 3688*m - 1)*x^10/10! + (1024*m^5 + 259328*m^4 + 1538560*m^3 + 870640*m^2 + 33212*m + 1)*x^12/12! + ...
dn(x,m) = 1 - m*x^2/2! + (m^2 + 4*m)*x^4/4! + (-m^3 - 44*m^2 - 16*m)*x^6/6! + (m^4 + 408*m^3 + 912*m^2 + 64*m)*x^8/8! + (-m^5 - 3688*m^4 - 30768*m^3 - 15808*m^2 - 256*m)*x^10/10! + (m^6 + 33212*m^5 + 870640*m^4 + 1538560*m^3 + 259328*m^2 + 1024*m)*x^12/12! + ...

Links

Crossrefs

Cf. A318005 (column 0).

Programs

  • PARI
    {T(n,k) = my(A=[1],S=x,C=1,D=1); for(i=0, n,
S = intformal(C*D +x*O(x^n));
C = 1 - intformal(S*D) ;
D = 1 - m*intformal(S*C); );
for(i=1,n, A=concat(A,0);
A[#A] = -Vec( subst(C,x,x*Ser(A) + x) + subst(S,x,x*Ser(A) - x) )[#A+1] );
n!*polcoeff(polcoeff(A,n,x),k,m)}
/* Print as a triangle: */
for(n=1,12,for(k=0,(n-1)\2,print1(T(n,k),", "));print(""))

Formula

E.g.f. A = A(x,m) = Sum_{n>=1} Sum_{k=0..floor((n-1)/2)} T(n,k)*x^n*m^k/n! satisfies:
(1) A(-A(-x, m), m) = x.
(2) 1 = cn(A + x, m) + sn(A - x, m).
(3) (cn(A) + sn(A)*dn(x)) * (cn(x) - sn(x)*dn(A)) = 1 - m*sn(x)^2*sn(A)^2, where parameter m is implicit.
Showing 1-3 of 3 results.

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