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 1-6 of 6 results.

A153302 G.f.: A(x) = cm4(x)^2 + sm4(x)^2 where cm4(x) and sm4(x) are the g.f.s of A153300 and A153301, respectively, that satisfy cm4(x)^4 - sm4(x)^4 = 1.

Original entry on oeis.org

1, 2, 12, 216, 7056, 368928, 28340928, 3000945024, 419025809664, 74600006164992, 16492933524114432, 4433180509950990336, 1423737921326106710016, 538417241668323364773888, 236818870322157143631249408
Offset: 0

Views

Author

Paul D. Hanna, Jan 02 2009

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x^2/2! + 12*x^4/4! + 216*x^6/6! + 7056*x^8/8! + 368928*x^10/10! + ...
From _Paul D. Hanna_, Apr 30 2009: (Start)
O.g.f.: G(x) = 1 + 2*x^2 + 12*x^4 + 216*x^6 + 7056*x^8 + ...
G(x) = 1/(1 - 2x^2/(1 - 4x^2/(1 - 18x^2/(1 - 16x^2/(1 - 50x^2/(1-...)))))).
(End)

References

  • E. van Fossen Conrad, Some continued fraction expansions of elliptic functions, PhD thesis, The Ohio State University, 2002, p. 35. [Paul Barry, Mar 29 2010]

Crossrefs

Cf. A153300, A153301.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, Boole[n == 0], With[ {m = 2 n}, m! SeriesCoefficient[ JacobiND[ x, 2], {x, 0, m}]]]; (* Michael Somos, Oct 18 2011 *)
a[ n_] := If[ n < 0, 0, With[{m = 2 n + 1}, (-1)^n m! SeriesCoefficient[ JacobiAmplitude[ x, 2], {x, 0, m}]]]; (* Michael Somos, Mar 13 2017 *)
Table[Abs[SeriesCoefficient[InverseSeries[Series[EllipticF[x, 2], {x, 0, 40}]],2 n + 1] (2 n + 1)!], {n, 0, 19}] (* Benedict W. J. Irwin, Apr 04 2017 *)
nmax = 20; s = CoefficientList[Series[JacobiNC[Sqrt[2] x, 1/2], {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!; Table[s[[2*n + 1]], {n, 0, nmax}] (* Vaclav Kotesovec, Nov 29 2020 *)
  • PARI
    {a(n) = my(A);if(n<0,0,A=x*O(x); for(i=0,n, A = 1 + intformal( intformal(A^3)^3 ) ); (2*n)!*polcoeff( A^2 + sqrt(A^4-1), 2*n))}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=1); A = deriv( serreverse( intformal( 1/sqrt(cosh(2*x + O(x^(2*n+2)))) ))); (2*n)!*polcoeff(A, 2*n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 13 2017

Formula

G.f. satisfies: A(x)*A(i*x) = 1 where A(x) = Sum_{n>=0} a(n)*x^(2n)/(2n)! and i^2=-1.
From Paul D. Hanna, Apr 30 2009: (Start)
The o.g.f. G(x), as the formal Laplace transform of e.g.f. cm4(x)^2 + sm4(x)^2, is given by the continued fraction:
G(x) = 1/(1-2(x)^2/(1-(2x)^2/(1-2(3x)^2/(1-(4x)^2/(1-2(5x)^2/(1-...)))))).
(End)
Let f(x) = sqrt(x^4-1). Let D be the operator f(x)*d/dx. Then it appears that D^(2*n-1)(f(x)) evaluated at x = 1 equals a(n) (checked up to a(14)). - Peter Bala, Aug 30 2011
G.f.: 1/Q(0), where Q(k)= 1 - 2*x*(2*k+1)^2/(1 - x*(2*k+2)^2/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
E.g.f.: A(x) = d/dx Series_Reversion( Integral sqrt( cosh(2*x) ) dx ). - Paul D. Hanna, Aug 13 2017

A153303 G.f.: cm4(x)^4 = Sum_{n>=0} a(n)*x^(4n)/(4n)!, where cm4(x) is defined by A153300.

Original entry on oeis.org

1, 24, 24192, 140507136, 2716743794688, 132091533948616704, 13574624941450494738432, 2619220630292562698311827456, 870703020893737265865222361448448
Offset: 0

Views

Author

Paul D. Hanna, Jan 03 2009

Keywords

Examples

			G.f.: cm4(x)^4 = 1 + 24*x^4/4! + 24192*x^8/8! + 140507136*x^12/12! +...
The functions:
cm4(x) = 1 + 6*x^4/4! + 2268*x^8/8! + 7434504*x^12/12! + 95227613712*x^16/16! +...
sm4(x) = x + 18*x^5/5! + 14364*x^9/9! + 70203672*x^13/13! + 1192064637456*x^17/17! +...
satisfy:
cm4(x)^4 - sm4(x)^4 = 1 ;
d/dx cm4(x) = sm4(x)^3 ;
d/dx sm4(x) = cm4(x)^3 .

Crossrefs

Cf. A153300 (cm4(x)), A153301 (sm4(x)), A153302 (cm4(x)^2+sm4(x)^2).

Programs

  • PARI
    {a(n)=local(A); if(n<0, 0, A=x*O(x); for(i=0, n, A=1+intformal(intformal(A^3)^3)); n=4*n; n!*polcoeff(A^4, n))}

Formula

Conjecture: a(n)/2^(4n-1) is an odd integer for n>0.

A153301 Coefficient of x^(4n+1)/(4n+1)! in the Maclaurin expansion of sm4(x), which is a generalization of the Dixon elliptic function sm(x,0) defined by A104133.

Original entry on oeis.org

1, 18, 14364, 70203672, 1192064637456, 52269828456672288, 4930307288899134335424, 884135650165992118901204352, 275721138550891190637445080842496
Offset: 0

Views

Author

Paul D. Hanna, Jan 02 2009

Keywords

Comments

Equals column 0 of triangle A357800.

Examples

			G.f.: sm4(x) = x + 18*x^5/5! + 14364*x^9/9! + 70203672*x^13/13! + 1192064637456*x^17/17! +...
cm4(x) = 1 + 6*x^4/4! + 2268*x^8/8! + 7434504*x^12/12! + 95227613712*x^16/16! +...
These functions satisfy: cm4(x)^4 - sm4(x)^4 = 1 where:
sm4(x)^4 = 24*x^4/4! + 24192*x^8/8! + 140507136*x^12/12! + 2716743794688*x^16/16! +...
RELATED EXPANSIONS:
sm4(x)^2 = 2*x^2/2! + 216*x^6/6! + 368928*x^10/10! + 3000945024*x^14/14! +...
cm4(x)^2 = 1 + 12*x^4/4! + 7056*x^8/8! + 28340928*x^12/12! + 419025809664*x^16/16! +...
sm4(x)^3 = 6*x^3/3! + 2268*x^7/7! + 7434504*x^11/11! + 95227613712*x^15/15! +...
cm4(x)^3 = 1 + 18*x^4/4! + 14364*x^8/8! + 70203672*x^12/12! + 1192064637456*x^16/16! +...
DERIVATIVES:
d/dx sm4(x) = cm4(x)^3 ;
d^2/dx^2 sm4(x) = 3*sm4(x)^3*cm4(x)^2 ;
d^3/dx^3 sm4(x) = 6*sm4(x)^6*cm4(x) + 9*sm4(x)^2*cm4(x)^5 ;
d^4/dx^4 sm4(x) = 6*sm4(x)^9 + 81*sm4(x)^5*cm4(x)^4 + 18*sm4(x)*cm4(x)^8 ;...

Links

Crossrefs

Cf. A104133; A153300, A153302 (cm4(x)^2 + sm4(x)^2).
Cf. A357800.

Programs

  • Mathematica
    With[{n = 8}, CoefficientList[Series[JacobiSN[Sqrt[2] x^(1/4), 1/2]/(x^(1/4) Sqrt[2 JacobiCN[Sqrt[2] x^(1/4), 1/2]]), {x, 0, n}], x] Table[(4 k + 1)!, {k, 0, n}]] (* Jan Mangaldan, Jan 04 2017 *)
  • PARI
    {a(n)=local(A);if(n<0,0,A=x*O(x);for(i=0,n,A=intformal((1+intformal(A^3))^3));n=4*n+1;n!*polcoeff(A,n))}

Formula

Define cm4(x)^4 = 1 + sm4(x)^4, where cm4(x) is the g.f. of A153300, then:
d/dx sm4(x) = cm4(x)^3 ;
d/dx cm4(x) = sm4(x)^3 .

A357801 Coefficients T(n,k) of x^(4*n)*r^(4*k)/(4*n)! in power series C(x,r) = 1 + Integral S(x,r)^3 * D(x,r)^3 dx such that C(x,r)^4 - S(x,r)^4 = 1 and D(x,r)^4 - r^4*S(x,r)^4 = 1, as a triangle read by rows.

Original entry on oeis.org

1, 6, 0, 2268, 6048, 0, 7434504, 56282688, 35126784, 0, 95227613712, 1409371197696, 2514356038656, 679185948672, 0, 3354162536029536, 81696140755536384, 284770675495950336, 220415417637617664, 33022883487154176, 0, 264444869673131894208, 9583398717725834749440, 54913653475645427527680, 83079959422282198548480, 35701050229143616880640, 3393656235362623684608, 0
Offset: 0

Views

Author

Paul D. Hanna, Oct 14 2022

Keywords

Comments

Equals a row reversal of triangle A357802.

Examples

			E.g.f.: C(x,r) = Sum_{n>=0} T(n,k) * x^(4*n) * r^(4*k) / (4*n)! begins:
C(x,r) = 1 + 6*x^4/4! + (2268 + 6048*r^4)*x^8/8! + (7434504 + 56282688*r^4 + 35126784*r^8)*x^12/12! + (95227613712 + 1409371197696*r^4 + 2514356038656*r^8 + 679185948672*r^12)*x^16/16! + (3354162536029536 + 81696140755536384*r^4 + 284770675495950336*r^8 + 220415417637617664*r^12 + 33022883487154176*r^16)*x^20/20! + (264444869673131894208 + 9583398717725834749440*r^4 + 54913653475645427527680*r^8 + 83079959422282198548480*r^12 + 35701050229143616880640*r^16 + 3393656235362623684608*r^20)*x^24/24! + ...
where S(x,r) = 1 + Integral S(x,r)^3 * D(x,r)^3 dx.
TRIANGLE.
This triangle of coefficients T(n,k) of x^(4*n) * r^(4*k) / (4*n)! in C(x,r) for n >= 0, k = 0..n, begins:
n = 0: [1];
n = 1: [6, 0];
n = 2: [2268, 6048, 0];
n = 3: [7434504, 56282688, 35126784, 0];
n = 4: [95227613712, 1409371197696, 2514356038656, 679185948672, 0];
n = 5: [3354162536029536, 81696140755536384, 284770675495950336, 220415417637617664, 33022883487154176, 0];
n = 6: [264444869673131894208, 9583398717725834749440, 54913653475645427527680, 83079959422282198548480, 35701050229143616880640, 3393656235362623684608, 0]; ...
in which column 0 equals A153300.
RELATED SERIES.
S(x,r) = x + (18 + 18*r^4)*x^5/5! + (14364 + 58968*r^4 + 14364*r^8)*x^9/9! + (70203672 + 671650056*r^4 + 671650056*r^8 + 70203672*r^12)*x^13/13! + (1192064637456 + 20707300240704*r^4 + 47530354598496*r^8 + 20707300240704*r^12 + 1192064637456*r^16)*x^17/17! + (52269828456672288 + 1437626817559769760*r^4 + 5941554215913771840*r^8 + 5941554215913771840*r^12 + 1437626817559769760*r^16 + 52269828456672288*r^20)*x^21/21! + (4930307288899134335424 + 197041019249105562351744*r^4 + 1283341580573615116868160*r^8 + 2308585363008068715943680*r^12 + 1283341580573615116868160*r^16 + 197041019249105562351744*r^20 + 4930307288899134335424*r^24)*x^25/25! + ...
where C(x,r)^4 - S(x,r)^4 = 1.
D(x,r) = 1 + 6*r^4*x^4/4! + (6048*r^4 + 2268*r^8)*x^8/8! + (35126784*r^4 + 56282688*r^8 + 7434504*r^12)*x^12/12! + (679185948672*r^4 + 2514356038656*r^8 + 1409371197696*r^12 + 95227613712*r^16)*x^16/16! + (33022883487154176*r^4 + 220415417637617664*r^8 + 284770675495950336*r^12 + 81696140755536384*r^16 + 3354162536029536*r^20)*x^20/20! + (3393656235362623684608*r^4 + 35701050229143616880640*r^8 + 83079959422282198548480*r^12 + 54913653475645427527680*r^16 + 9583398717725834749440*r^20 + 264444869673131894208*r^24)*x^24/24! +
where D(x,r)^4 - r^4 * C(x,r)^4 = 1 - r^4.

Links

Crossrefs

Cf. A153300 (column 0), A357805 (row sums), A357800 (S(x,r)), A357802 (D(x,r)).
Cf. A357541.

Programs

  • PARI
    {T(n, k) = my(S=x, C=1, D=1); for(i=0, n,
S = intformal( C^3*D^3 +O(x^(4*n+4)));
C = 1 + intformal( S^3*D^3);
D = 1 + r^4*intformal( S^3*C^3); );
(4*n)!*polcoeff( polcoeff(C, 4*n, x), 4*k, r)}
for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))
  • PARI
    /* Using Series Reversion (faster) */
{T(n, k) = my(S = serreverse( intformal( 1/((1 + x^4)^3*(1 + r^4*x^4)^3 +O(x^(4*n+4)) )^(1/4) )) );
(4*n)!*polcoeff( polcoeff( (1 + S^4)^(1/4), 4*n, x), 4*k, r)}
for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))

Formula

Generating function C(x,r) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(4*n) * r^(4*k) / (4*n)! and related functions S(x,r) and D(x,r) satisfy the following formulas.
For brevity, some formulas here will use S = S(x,r), C = C(x,r), and D = D(x,r).
(1.a) C(x,r)^4 - S(x,r)^4 = 1.
(1.b) D(x,r)^4 - r^4 * S(x,r)^4 = 1.
(1.c) D(x,r)^4 - r^4 * C(x,r)^4 = 1 - r^4.
Integral formulas.
(2.a) S(x,r) = Integral C(x,r)^3 * D(x,r)^3 dx.
(2.b) C(x,r) = 1 + Integral S(x,r)^3 * D(x,r)^3 dx.
(2.c) D(x,r) = 1 + r^4 * Integral S(x,r)^3 * C(x,r)^3 dx.
(2.d) C(x,r)^4 = 1 + 4 * Integral S(x,r)^3 * C(x,r)^3 * D(x,r)^3 dx.
(2.e) D(x,r)^4 = 1 + 4*r^4 * Integral S(x,r)^3 * C(x,r)^3 * D(x,r)^3 dx.
Derivatives.
(3.a) d/dx S(x,r) = C(x,r)^3 * D(x,r)^3.
(3.b) d/dx C(x,r) = S(x,r)^3 * D(x,r)^3.
(3.c) d/dx D(x,r) = r^4 * S(x,r)^3 * C(x,r)^3.
Exponential formulas.
(4.a) C + S = exp( Integral (C^2 - C*S + S^2) * D^3 dx ).
(4.b) D + r*S = exp( r * Integral (D^2 - r*D*S + r^2*S^2) * C^3 dx ).
(4.c) C - S = exp( -Integral (C^2 + C*S + S^2) * D^3 dx ).
(4.d) D - r*S = exp( -r * Integral (D^2 + r*D*S + r^2*S^2) * C^3 dx ).
(5.a) C^2 + S^2 = exp( 2 * Integral S*C * D^3 dx ).
(5.b) D^2 + r^2*S^2 = exp( 2*r^2 * Integral S*D * C^3 dx ).
(5.c) C^2 - S^2 = exp( -2 * Integral S*C * D^3 dx ).
(5.d) D^2 - r^2*S^2 = exp( -2*r^2 * Integral S*D * C^3 dx ).
Hyperbolic functions.
(6.a) C = sqrt(C^2 - S^2) * cosh( Integral (C^2 + S^2) * D^3 dx ).
(6.b) S = sqrt(C^2 - S^2) * sinh( Integral (C^2 + S^2) * D^3 dx ).
(6.c) D = sqrt(D^2 - r^2*S^2) * cosh( r * Integral (D^2 + r^2*S^2) * C^3 dx ).
(6.d) r*S = sqrt(D^2 - r^2*S^2) * sinh( r * Integral (D^2 + r^2*S^2) * C^3 dx ).
(7.a) C^2 = cosh( 2 * Integral S*C * D^3 dx ).
(7.b) S^2 = sinh( 2 * Integral S*C * D^3 dx ).
(7.c) D^2 = cosh( 2*r^2 * Integral S*D * C^3 dx ).
(7.d) r^2*S^2 = sinh( 2*r^2 * Integral S*D * C^3 dx ).
Other formulas.
(8) S(x,r) = Series_Reversion( Integral 1/((1 + x^4)*(1 + r^4*x^4))^(3/4) dx ).
(9.a) T(n,0) = T(n,n) = A153301(n).
(9.b) Sum_{k=0..n} T(n,k) = A357805(n), for n >= 0.
From Paul D. Hanna, Apr 12 2023 (Start):
Let F(x,r) = Integral 1/((1 + x^4)*(1 + r^4*x^4))^(3/4) dx, then
(10.a) S( F(x,r), r) = x,
(10.b) C( F(x,r), r) = (1 + x^4)^(1/4),
(10.c) D( F(x,r), r) = (1 + r^4*x^4)^(1/4). (End)

A357802 Coefficients T(n,k) of x^(4*n)*r^(4*k)/(4*n)! in power series D(x,r) = 1 + r^4 * Integral S(x,r)^3 * C(x,r)^3 dx such that C(x,r)^4 - S(x,r)^4 = 1 and D(x,r)^4 - r^4*S(x,r)^4 = 1, as a triangle read by rows.

Original entry on oeis.org

1, 0, 6, 0, 6048, 2268, 0, 35126784, 56282688, 7434504, 0, 679185948672, 2514356038656, 1409371197696, 95227613712, 0, 33022883487154176, 220415417637617664, 284770675495950336, 81696140755536384, 3354162536029536, 0, 3393656235362623684608, 35701050229143616880640, 83079959422282198548480, 54913653475645427527680, 9583398717725834749440, 264444869673131894208
Offset: 0

Views

Author

Paul D. Hanna, Oct 14 2022

Keywords

Comments

Equals a row reversal of triangle A357801.

Examples

			E.g.f.: D(x,r) = Sum_{n>=0} T(n,k) * x^(4*n) * r^(4*k) / (4*n)! begins:
D(x,r) = 1 + 6*r^4*x^4/4! + (6048*r^4 + 2268*r^8)*x^8/8! + (35126784*r^4 + 56282688*r^8 + 7434504*r^12)*x^12/12! + (679185948672*r^4 + 2514356038656*r^8 + 1409371197696*r^12 + 95227613712*r^16)*x^16/16! + (33022883487154176*r^4 + 220415417637617664*r^8 + 284770675495950336*r^12 + 81696140755536384*r^16 + 3354162536029536*r^20)*x^20/20! + (3393656235362623684608*r^4 + 35701050229143616880640*r^8 + 83079959422282198548480*r^12 + 54913653475645427527680*r^16 + 9583398717725834749440*r^20 + 264444869673131894208*r^24)*x^24/24! +
where D(x,r) = 1 + r^4 * Integral S(x,r)^3 * D(x,r)^3 dx.
TRIANGLE.
This triangle of coefficients T(n,k) of x^(4*n) * r^(4*k) / (4*n)! in D(x,r) for n >= 0, k = 0..n, begins:
n = 0: [1];
n = 1: [0, 6];
n = 2: [0, 6048, 2268];
n = 3: [0, 35126784, 56282688, 7434504];
n = 4: [0, 679185948672, 2514356038656, 1409371197696, 95227613712];
n = 5: [0, 33022883487154176, 220415417637617664, 284770675495950336, 81696140755536384, 3354162536029536];
n = 6: [0, 3393656235362623684608, 35701050229143616880640, 83079959422282198548480, 54913653475645427527680, 9583398717725834749440, 264444869673131894208]; ...
in which the main diagonal equals A153300.
RELATED SERIES.
S(x,r) = x + (18 + 18*r^4)*x^5/5! + (14364 + 58968*r^4 + 14364*r^8)*x^9/9! + (70203672 + 671650056*r^4 + 671650056*r^8 + 70203672*r^12)*x^13/13! + (1192064637456 + 20707300240704*r^4 + 47530354598496*r^8 + 20707300240704*r^12 + 1192064637456*r^16)*x^17/17! + (52269828456672288 + 1437626817559769760*r^4 + 5941554215913771840*r^8 + 5941554215913771840*r^12 + 1437626817559769760*r^16 + 52269828456672288*r^20)*x^21/21! + (4930307288899134335424 + 197041019249105562351744*r^4 + 1283341580573615116868160*r^8 + 2308585363008068715943680*r^12 + 1283341580573615116868160*r^16 + 197041019249105562351744*r^20 + 4930307288899134335424*r^24)*x^25/25! + ...
where D(x,r)^4 - r^4*S(x,r)^4 = 1.
C(x,r) = 1 + 6*x^4/4! + (2268 + 6048*r^4)*x^8/8! + (7434504 + 56282688*r^4 + 35126784*r^8)*x^12/12! + (95227613712 + 1409371197696*r^4 + 2514356038656*r^8 + 679185948672*r^12)*x^16/16! + (3354162536029536 + 81696140755536384*r^4 + 284770675495950336*r^8 + 220415417637617664*r^12 + 33022883487154176*r^16)*x^20/20! + (264444869673131894208 + 9583398717725834749440*r^4 + 54913653475645427527680*r^8 + 83079959422282198548480*r^12 + 35701050229143616880640*r^16 + 3393656235362623684608*r^20)*x^24/24! + ...
where D(x,r)^4 - r^4 * C(x,r)^4 = 1 - r^4.

Links

Crossrefs

Cf. A153300 (diagonal), A357805 (row sums), A357800 (S(x,r)), A357801 (C(x,r)).
Cf. A357542.

Programs

  • PARI
    {T(n, k) = my(S=x, C=1, D=1); for(i=0, n,
S = intformal( C^3*D^3 +O(x^(4*n+4)));
C = 1 + intformal( S^3*D^3);
D = 1 + r^4*intformal( S^3*C^3); );
(4*n)!*polcoeff( polcoeff(D, 4*n, x), 4*k, r)}
for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))
  • PARI
    /* Using Series Reversion (faster) */
{T(n, k) = my(S = serreverse( intformal( 1/((1 + x^4)^3*(1 + r^4*x^4)^3 +O(x^(4*n+4)) )^(1/4) )) );
(4*n)!*polcoeff( polcoeff( (1 + r^4*S^4)^(1/4), 4*n, x), 4*k, r)}
for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))

Formula

Generating function D(x,r) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(4*n) * r^(4*k) / (4*n)! and related functions S(x,r) and C(x,r) satisfy the following formulas.
For brevity, some formulas here will use S = S(x,r), C = C(x,r), and D = D(x,r).
(1.a) C(x,r)^4 - S(x,r)^4 = 1.
(1.b) D(x,r)^4 - r^4 * S(x,r)^4 = 1.
(1.c) D(x,r)^4 - r^4 * C(x,r)^4 = 1 - r^4.
Integral formulas.
(2.a) S(x,r) = Integral C(x,r)^3 * D(x,r)^3 dx.
(2.b) C(x,r) = 1 + Integral S(x,r)^3 * D(x,r)^3 dx.
(2.c) D(x,r) = 1 + r^4 * Integral S(x,r)^3 * C(x,r)^3 dx.
(2.d) D(x,r)^4 = 1 + r^4 * Integral 4 * S(x,r)^3 * C(x,r)^3 * D(x,r)^3 dx.
Derivatives.
(3.a) d/dx S(x,r) = C(x,r)^3 * D(x,r)^3.
(3.b) d/dx C(x,r) = S(x,r)^3 * D(x,r)^3.
(3.c) d/dx D(x,r) = r^4 * S(x,r)^3 * C(x,r)^3.
Exponential formulas.
(4.a) C + S = exp( Integral (C^2 - C*S + S^2) * D^3 dx ).
(4.b) D + r*S = exp( r * Integral (D^2 - r*D*S + r^2*S^2) * C^3 dx ).
(4.c) C - S = exp( -Integral (C^2 + C*S + S^2) * D^3 dx ).
(4.d) D - r*S = exp( -r * Integral (D^2 + r*D*S + r^2*S^2) * C^3 dx ).
(5.a) C^2 + S^2 = exp( 2 * Integral S*C * D^3 dx ).
(5.b) D^2 + r^2*S^2 = exp( 2*r^2 * Integral S*D * C^3 dx ).
(5.c) C^2 - S^2 = exp( -2 * Integral S*C * D^3 dx ).
(5.d) D^2 - r^2*S^2 = exp( -2*r^2 * Integral S*D * C^3 dx ).
Hyperbolic functions.
(6.a) C = sqrt(C^2 - S^2) * cosh( Integral (C^2 + S^2) * D^3 dx ).
(6.b) S = sqrt(C^2 - S^2) * sinh( Integral (C^2 + S^2) * D^3 dx ).
(6.c) D = sqrt(D^2 - r^2*S^2) * cosh( r * Integral (D^2 + r^2*S^2) * C^3 dx ).
(6.d) r*S = sqrt(D^2 - r^2*S^2) * sinh( r * Integral (D^2 + r^2*S^2) * C^3 dx ).
(7.a) C^2 = cosh( 2 * Integral S*C * D^3 dx ).
(7.b) S^2 = sinh( 2 * Integral S*C * D^3 dx ).
(7.c) D^2 = cosh( 2*r^2 * Integral S*D * C^3 dx ).
(7.d) r^2*S^2 = sinh( 2*r^2 * Integral S*D * C^3 dx ).
Other formulas.
(8) S(x,r) = Series_Reversion( Integral 1/((1 + x^4)*(1 + r^4*x^4))^(3/4) dx ).
(9.a) T(n,0) = T(n,n) = A153301(n).
(9.b) Sum_{k=0..n} T(n,k) = A357805(n), for n >= 0.
From Paul D. Hanna, Apr 12 2023 (Start):
Let F(x,r) = Integral 1/((1 + x^4)*(1 + r^4*x^4))^(3/4) dx, then
(10.a) S( F(x,r), r) = x,
(10.b) C( F(x,r), r) = (1 + x^4)^(1/4),
(10.c) D( F(x,r), r) = (1 + r^4*x^4)^(1/4). (End)

A357805 a(n) = coefficient of x^(4*n)/(4*n)! in power series C(x) = 1 + Integral S(x)^3 * C(x)^3 dx such that C(x)^4 - S(x)^4 = 1.

Original entry on oeis.org

1, 6, 8316, 98843976, 4698140798736, 623259279912288096, 186936162949832833285056, 110352751044119383032310847616, 116215132158682166284921510741483776, 202905498509713715271588290261091671041536, 554890365215965228675768455367962915432839248896
Offset: 0

Views

Author

Paul D. Hanna, Oct 14 2022

Keywords

Comments

Equals row sums of triangles A357801 and A357802.

Examples

			E.g.f.: C(x) = 1 + 6*x^4/4! + 8316*x^8/8! + 98843976*x^12/12! + 4698140798736*x^16/16! + 623259279912288096*x^20/20! + 186936162949832833285056*x^24/24! + 110352751044119383032310847616*x^28/28! + ...
such that
C( Integral 1/(1 + x^4)^(3/2) dx ) = (1 + x^4)^(1/4)
also
C(x)^4 - S(x)^4 = 1,
where
S(x) = x + 36*x^5/5! + 87696*x^9/9! + 1483707456*x^13/13! + 91329084354816*x^17/17! + 14862901723860427776*x^21/21! + 5279211177231308343054336*x^25/25! + ... + A357804(n)*x^(4*n+1)/(4*n+1)! + ...

Links

Crossrefs

Cf. A357804 (S(x)), A357801, A357802, A153300.

Programs

  • PARI
    /* Using Series Reversion (faster) */
{a(n) = my(S = serreverse( intformal( 1/(1 + x^4 +O(x^(4*n+4)))^(3/2) )) );
(4*n)!*polcoeff( (1 + S^4)^(1/4), 4*n)}
for(n=0, 10, print1( a(n), ", "))
  • PARI
    {a(n) = my(S=x, C=1); for(i=0, n,
S = intformal( C^6 +O(x^(4*n+4)));
C = 1 + intformal( S^3*C^3 ) );
(4*n)!*polcoeff( C, 4*n)}
for(n=0, 10, print1( a(n), ", "))

Formula

Generating function C(x) = Sum_{n>=0} a(n)*x^(4*n)/(4*n)! and related function S(x) satisfies the following formulas.
For brevity, some formulas here will use C = C(x) and S = S(x), where S(x) = (C(x)^4 - 1)^(1/4) is the e.g.f. of A357804.
(1) C(x)^4 - S(x)^4 = 1.
Integral formulas.
(2.a) S(x) = Integral C(x)^6 dx.
(2.b) C(x) = 1 + Integral S(x)^3 * C(x)^3 dx.
(2.c) S(x)^4 = Integral 4 * S(x)^3 * C(x)^6 dx.
(2.d) C(x)^4 = 1 + Integral 4 * S(x)^3 * C(x)^6 dx.
Derivatives.
(3.a) d/dx S(x) = C(x)^6.
(3.b) d/dx C(x) = S(x)^3 * C(x)^3.
Exponential formulas.
(4.a) C + S = exp( Integral (C^2 - C*S + S^2) * C^3 dx ).
(4.b) C - S = exp( -Integral (C^2 + C*S + S^2) * C^3 dx ).
(5.a) C^2 + S^2 = exp( 2 * Integral S*C^4 dx ).
(5.b) C^2 - S^2 = exp( -2 * Integral S*C^4 dx ).
Hyperbolic functions.
(6.a) C = sqrt(C^2 - S^2) * cosh( Integral (C^2 + S^2) * C^3 dx ).
(6.b) S = sqrt(C^2 - S^2) * sinh( Integral (C^2 + S^2) * C^3 dx ).
(7.a) C^2 = cosh( 2 * Integral S*C^4 dx ).
(7.b) S^2 = sinh( 2 * Integral S*C^4 dx ).
Explicit formulas.
(8.a) S(x) = Series_Reversion( Integral 1/(1 + x^4)^(3/2) dx ).
(8.b) C( Integral 1/(1 + x^4)^(3/2) dx ) = (1 + x^4)^(1/4).
Showing 1-6 of 6 results.

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