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.

A370542 Expansion of the Jacobi elliptic function sn(x,k) at k = 2 (odd powers only).

Original entry on oeis.org

1, -5, 73, -2765, 171409, -16145045, 2168436697, -391723265885, 91633164775201, -26955095234906405, 9737498127795037033, -4237907290209405609965, 2187044171819241257792689, -1320533769141977996485790645, 922274662722967857470247551737, -737730926392606318468533810754685
Offset: 0

Views

Author

Paul D. Hanna, Mar 23 2024

Keywords

Examples

			E.g.f.: S(x) = x - 5*x^3/3! + 73*x^5/5! - 2765*x^7/7! + 171409*x^9/9! - 16145045*x^11/11! + 2168436697*x^13/13! - 391723265885*x^15/15! + ...
where S(x) = sn(x,2).

References

  • H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 374.

Links

Crossrefs

Cf. A000182 (unsigned sn(x,1)), A060628 (sn(x,k)).
Cf. A370543 (cn(x,2)), A370544 (dn(x,2)), A249282.

Programs

  • Maple
    # a(n) = (2*n+1)! * [x^(2*n+1)] sn(x, 2).
sn_list := proc(k, len) local n; seq((2*n+1)!*coeff(series(JacobiSN(z, k), z,
2*len + 2), z, 2*n + 1), n = 0..len) end:
sn_list(2, 15);  # Peter Luschny, Mar 25 2024
  • Mathematica
    nmax = 20;
DeleteCases[CoefficientList[JacobiSN[x, 4] + O[x]^(2*nmax+2), x], 0]* (2*Range[0, nmax] + 1)! (* Jean-François Alcover, Mar 28 2024 *)
  • PARI
    /* S(x) = Jacobi Elliptic Function sn(x,k) at k = 2: */
{a(n) = my(S, k = 2); S = serreverse( intformal( 1/sqrt((1-x^2)*(1-k^2*x^2 +x*O(x^(2*n+2)) ) ) ));
(2*n+1)!*polcoeff(S,2*n+1)}
for(n=0,20, print1( a(n), ", ") );

Formula

a(n) = (-1)^n * Sum_{k=0..n} A060628(n,k)*4^k for n >= 0.
E.g.f. S(x) = Sum_{n>=0} a(n)*x^(2*n+1)/(2*n+1)! satisfies the following formulas, where sn, cn, and dn are Jacobi elliptic functions.
(1) S(x) = sn(x,k) at k = 2.
(2.a) S(x) = sn(2*x,1/2)/2.
(2.b) S(x) = sn(x,1/2) * cn(x,1/2) * dn(x,1/2) / (1 - sn(x,1/2)^4/4).
(3.a) S(x) = Series_Reversion( Integral 1/sqrt( (1-x^2)*(1-4*x^2) ) dx ).
(3.b) S(x) = Integral sqrt(1 - S(x)^2) * sqrt(1 - 4*S(x)^2) dx.
(4.a) S(x) = sin( Integral sqrt(1 - 4*S(x)^2) dx ).
(4.b) S(x) = sin( 2 * Integral sqrt(1 - S(x)^2) dx ) / 2.
(5.a) S(x) = sqrt(1 - cn(x,2)^2).
(5.b) S(x) = sqrt(1 - dn(x,2)^2) / 2.
O.g.f.: x/(1 + 5*x - 4*1*2^2*3*x^2/(1 + 5*3^2*x - 4*3*4^2*5*x^2/(1 + 5*5^2*x - 4*5*6^2*7*x^2/(1 + 5*7^2*x - 4*7*8^2*9*x^2/(1 + 5*9^2*x - ...))))) = x - 5*x^2 + 73*x^3 - 2765*x^4 + 171409*x^5 - 16145045*x^6 + ... (continued fraction, see Wall, 94.17, p. 374).
a(n) ~ (-1)^n * 2^(4*n+4) * agm(1,2)^(2*n+2) * n^(2*n + 3/2) / (Pi^(2*n + 3/2) * exp(2*n)), where agm(1,2) = A068521 is the arithmetic-geometric mean. - Vaclav Kotesovec, Mar 28 2024

A370544 Expansion of the Jacobi elliptic function dn(x,k) at k = 2 (even powers only).

Original entry on oeis.org

1, -4, 32, -832, 41216, -3168256, 359518208, -56319950848, 11624409595904, -3059387770077184, 999955757611876352, -397353151288859164672, 188655750511199441125376, -105472284295853235792510976, 68582751548430569936978444288, -51320267059211655419226235076608
Offset: 0

Views

Author

Paul D. Hanna, Mar 25 2024

Keywords

Examples

			E.g.f.: D(x) = 1 - 4*x^2/2! + 32*x^4/4! - 832*x^6/6! + 41216*x^8/8! - 3168256*x^10/10! + 359518208*x^12/12! - 56319950848*x^14/14! + ...
where D(x) = dn(x,2).

References

  • H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 374.

Links

Crossrefs

Cf. A028296 (dn(x,1)), A060627 (cn(x,k)).
Cf. A370542 (sn(x,2)), A370543 (cn(x,2)), A249282.

Programs

  • Maple
    # a(n) = (2*n)! * [x^(2*n)] dn(x, 2).
dn_list := proc(k, len) local n; seq((2*n)!*coeff(series(JacobiDN(z, k), z,
2*len + 2), z, 2*n), n = 0..len) end:
dn_list(2, 15);  # Peter Luschny, Mar 25 2024
  • Mathematica
    nmax = 20;
DeleteCases[CoefficientList[JacobiDN[x, 4] + O[x]^(2*nmax+2), x], 0]* (2*Range[0, nmax])! (* Jean-François Alcover, Mar 28 2024 *)
  • PARI
    /* D(x) = Jacobi Elliptic Function dn(x,k) at k = 2: */
{a(n) = my(k=2, C=1,S=x,D=1); for(i=1,n,
S = intformal(C*D + x*O(x^(2*n+1)));
C = 1 - intformal(S*D);
D = 1 - k^2*intformal(S*C)); (2*n)!*polcoeff(D,2*n)}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = (-1)^n * Sum_{k=0..n-1} A060627(n,k)*4^(n-k) for n >= 1, with a(0) = 1.
E.g.f. D(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! satisfies the following formulas, where sn, cn, and dn are Jacobi elliptic functions.
(1) D(x) = dn(x,k) at k = 2.
(2.a) D(x) = cn(2*x, 1/2).
(2.b) D(x) = (4 - 8*sn(x,1/2)^2 + sn(x,1/2)^4) / (4 - sn(x,1/2)^4).
(3) D(x) = 1 - Integral sqrt(1 - D(x)^2) * sqrt(3 + D(x)^2) dx.
(4) D(x) = cos( Integral sqrt(3 + D(x)^2) dx ).
(5.a) D(x) = sqrt(1 - 4*sn(x,2)^2).
(5.b) D(x) = sqrt(4*cn(x,2)^2 - 3).
O.g.f. 1/(1 + 4*x/(1 + 2^2*x/(1 + 4*3^2*x/(1 + 4^2*x/(1 + 4*5^2*x/(1 + 6^2*x/(1 + 4*7^2*x/(1 + ...)))))))) = 1 - 4*x + 32*x^2 - 832*x^3 + 41216*x^4 - 3168256*x^5 + ... (continued fraction, see Wall, 94.19, p. 374).
a(n) ~ (-1)^n * 2^(4*n+3) * agm(1,2)^(2*n+1) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)), where agm(1,2) = A068521 is the arithmetic-geometric mean. - Vaclav Kotesovec, Mar 28 2024

A371055 Ternary numbers consisting of a run of 1's, then a run of 0's, then a run of 2's.

Original entry on oeis.org

102, 1002, 1022, 1102, 10002, 10022, 10222, 11002, 11022, 11102, 100002, 100022, 100222, 102222, 110002, 110022, 110222, 111002, 111022, 111102, 1000002, 1000022, 1000222, 1002222, 1022222, 1100002, 1100022, 1100222, 1102222, 1110002, 1110022, 1110222
Offset: 1

Views

Author

Clark Kimberling, Mar 22 2024

Keywords

Examples

			10002 consists of a run 1, then a run 000, then a run 2.

Crossrefs

Cf. A007089, A371054

Programs

  • Mathematica
    Map[#[[1]] &, Select[Map[{#, Map[#[[1]] &, Split[IntegerDigits[#, 3]]] == {1, 0, 2}} &, Range[2, 4000, 3]], #[[2]] &]]  (* A370543 *)
ToExpression[Map[IntegerString[#, 3] &, %]]  (* this sequence *)
(* Peter J. C. Moses, Mar 06 2024 *)
  • Python
    from itertools import count, islice
def A371055_gen(): # generator of terms
    return (10**(l-a)*((10**a-1)//9)+((10**b-1)//9<<1) for l in count(3) for a in range(1,l-1) for b in range(1,l-a))
A371055_list = list(islice(A371055_gen(),20)) # Chai Wah Wu, Mar 25 2024
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.