A153302 -id:A153302 - cp's OEIS frontend

A153300 Coefficient of x^(4n)/(4n)! in the Maclaurin expansion of cm4(x), which is a generalization of the Dixon elliptic function cm(x,0) defined by A104134.

1, 6, 2268, 7434504, 95227613712, 3354162536029536, 264444869673131894208, 40740588107524550752746624, 11136881432872615930509713801472, 5026062205760019668688216299061782016

Offset: 0

Paul D. Hanna, Jan 02 2009

nonn

Equals column 0 of triangle A357801.

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..117
Alessandro Gambini, Giorgio Nicoletti, and Daniele Ritelli, The Wallis Products for Fermat Curves, Vietnam J. Math. (2023).
Alessandro Gambini, Giorgio Nicoletti, and Daniele Ritelli, A Structural Approach to Gudermannian Functions, Results in Mathematics (2024) Vol. 79, Art. No. 10.

Cf. A104134; A153301, A153302 (cm4(x)^2 + sm4(x)^2).

Cf. A153303 (cm4(x)^4), A357801.

With[{n = 9}, CoefficientList[Series[JacobiDN[Sqrt[2] x^(1/4), 1/2]/Sqrt[JacobiCN[Sqrt[2] x^(1/4), 1/2]], {x, 0, n}], x] Table[(4 k)!, {k, 0, n}]] (* Jan Mangaldan, Jan 04 2017 *)

{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,n))}

Define sm4(x)^4 = cm4(x)^4 - 1, where sm4(x) is the g.f. of A153301, then:
d/dx cm4(x) = sm4(x)^3 ;
d/dx sm4(x) = cm4(x)^3 .
a(n) ~ 2^(14*n + 11/4) * Gamma(3/4)^(8*n+1) * n^(4*n) / (exp(4*n) * Pi^(6*n + 3/4)). - Vaclav Kotesovec, Oct 06 2019

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

Paul D. Hanna, Jan 02 2009

nonn

Equals column 0 of triangle A357800.

			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 ;...

Alessandro Gambini, Giorgio Nicoletti, and Daniele Ritelli, The Wallis Products for Fermat Curves, Vietnam J. Math. (2023).
Alessandro Gambini, Giorgio Nicoletti, and Daniele Ritelli, A Structural Approach to Gudermannian Functions, Results in Mathematics (2024) Vol. 79, Art. No. 10.

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

Cf. A357800.

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 *)

{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))}

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 .

A290879 E.g.f. L(x) of aerated sequence satisfies: L(x) = Integral 1 / sqrt( cosh(2*L(x)) ) dx.

Original entry on oeis.org

1, -2, 44, -2840, 367760, -79719200, 26016555200, -11921650083200, 7300922254496000, -5759532173685056000, 5688335131502291840000, -6875441991877610827520000, 9983390897443366347676160000, -17148197258942716354314368000000, 34392433372153876998446324480000000, -79646680171456811546338888517120000000, 210930739548407111241584046599398400000000

Offset: 1

Paul D. Hanna, Aug 13 2017

sign

			E.g.f.: L(x) = x - 2*x^3/3! + 44*x^5/5! - 2840*x^7/7! + 367760*x^9/9! - 79719200*x^11/11! + 26016555200*x^13/13! - 11921650083200*x^15/15! + 7300922254496000*x^17/17! - 5759532173685056000*x^19/19! + 5688335131502291840000*x^21/21! +...
where
1/sqrt( cosh(2*L(x)) ) = 1 - 2*x^2/2! + 44*x^4/4! - 2840*x^6/6! + 367760*x^8/8! - 79719200*x^10/10! + 26016555200*x^12/12! - 11921650083200*x^14/14! +...

Robert Israel, Table of n, a(n) for n = 1..200

Cf. A290880, A290881, A290882, A290883, A153302.

Q:= series(Int(sqrt(cosh(2*t)),t),t,100):
S:= series(RootOf(Q-y,t),y,100):
seq(coeff(S,y,j)*j!,j=1..100,2); # Robert Israel, Aug 13 2017

terms = 17; m = 2 terms; cc = CoefficientList[InverseSeries[Integrate[Sqrt[ Cosh[2 x]] + O[x]^m, x], x], x]; DeleteCases[cc * Range[0, m-1]!, 0] (* Jean-François Alcover, Apr 02 2019 *)

{a(n) = my(L=x); for(i=1,n, L = intformal( 1/sqrt(cosh(2*L + O(x^(2*n+2)))) )); (2*n-1)!*polcoeff(L,2*n-1)}
for(n=1,20, print1(a(n),", "))

{a(n) = my(L=x); L = serreverse( intformal( sqrt(cosh(2*x + O(x^(2*n+2)))) )); (2*n-1)!*polcoeff(L,2*n-1)}
for(n=1,20, print1(a(n),", "))

E.g.f.: L(x) = Series_Reversion( Integral sqrt( cosh(2*x) ) dx ).
Let C(x) and S(x) be the e.g.f.s of A290880 and A290881, respectively, then e.g.f. L(x) satisfies:
(1) L(x) = log(C(x) + S(x)),
(2) cosh(2*L(x)) = C(x)^2 + S(x)^2,
(3) cosh(L(x)) = C(x) and sinh(L(x)) = S(x),
where C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1.

A290880 E.g.f. C(x) satisfies: C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1, where S(x) is the e.g.f. of A290881.

Original entry on oeis.org

1, 1, -7, 265, -24175, 4037425, -1070526775, 412826556025, -218150106913375, 151297155973926625, -133288452772763494375, 145378048431548466795625, -192296944484564858674279375, 303266384253858232005535140625, -562167814015907092875287424484375, 1210147640238238850996978598797265625, -2993757681527630470101347134338702109375

Offset: 0

Paul D. Hanna, Aug 13 2017

sign

			E.g.f.: C(x) = 1 + x^2/2! - 7*x^4/4! + 265*x^6/6! - 24175*x^8/8! + 4037425*x^10/10! - 1070526775*x^12/12! + 412826556025*x^14/14! - 218150106913375*x^16/16! + 151297155973926625*x^18/18! - 133288452772763494375*x^20/20! +...
such that C(x)^2 - S(x)^2 = 1 where S(x) begins:
S(x) = x - x^3/3! + 25*x^5/5! - 1705*x^7/7! + 227665*x^9/9! - 50333425*x^11/11! + 16655398825*x^13/13! - 7711225809625*x^15/15! + 4760499335502625*x^17/17! - 3779764853639958625*x^19/19! + 3752942823715824285625*x^21/21! +...

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A290879, A290881, A290882, A290883, A153302.

{a(n) = my(C=1,S=x); for(i=1,n, C = 1 + intformal( S/sqrt(C^2 + S^2 + O(x^(2*n+2))) ); S = intformal( C/sqrt(C^2 + S^2)) ); (2*n)!*polcoeff(C,2*n)}
for(n=0,20, print1(a(n),", "))

{a(n) = my(C=1); C = cosh( serreverse( intformal( sqrt(cosh(2*x + O(x^(2*n+2)))) ) )); (2*n)!*polcoeff(C,2*n)}
for(n=0,20, print1(a(n),", "))

E.g.f.: C(x) = cosh( Series_Reversion( Integral sqrt( cosh(2*x) ) dx ) ).
Let S(x) be the e.g.f. of A290881, then:
(1) C'(x) = S(x) / sqrt(C(x)^2 + S(x)^2).
(2) S'(x) = C(x) / sqrt(C(x)^2 + S(x)^2).
such that C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1.

A290881 E.g.f. S(x) satisfies: C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1, where C(x) is the e.g.f. of A290880.

Original entry on oeis.org

1, -1, 25, -1705, 227665, -50333425, 16655398825, -7711225809625, 4760499335502625, -3779764853639958625, 3752942823715824285625, -4556465805050372544735625, 6641455313355871353308640625, -11445605320939175012746492140625, 23021828780691053491298409381015625, -53450977127256739279274500814544765625

Offset: 1

Paul D. Hanna, Aug 13 2017

sign

			E.g.f.: S(x) = x - x^3/3! + 25*x^5/5! - 1705*x^7/7! + 227665*x^9/9! - 50333425*x^11/11! + 16655398825*x^13/13! - 7711225809625*x^15/15! + 4760499335502625*x^17/17! - 3779764853639958625*x^19/19! + 3752942823715824285625*x^21/21! +...
such that C(x)^2 - S(x)^2 = 1 where C(x) begins:
C(x) = 1 + x^2/2! - 7*x^4/4! + 265*x^6/6! - 24175*x^8/8! + 4037425*x^10/10! - 1070526775*x^12/12! + 412826556025*x^14/14! - 218150106913375*x^16/16! + 151297155973926625*x^18/18! - 133288452772763494375*x^20/20! +...

Paul D. Hanna, Table of n, a(n) for n = 1..100

Cf. A290879, A290880, A290882, A290883, A153302.

{a(n) = my(C=1,S=x); for(i=1,n, C = 1 + intformal( S/sqrt(C^2 + S^2 + O(x^(2*n+2))) ); S = intformal( C/sqrt(C^2 + S^2)) ); (2*n-1)!*polcoeff(S,2*n-1)}
for(n=1,20, print1(a(n),", "))

{a(n) = my(C=1); S = serreverse( intformal( sqrt( (1+2*x^2) / (1+x^2 + O(x^(2*n+2))) ) )); (2*n-1)!*polcoeff(S,2*n-1)}
for(n=1,20, print1(a(n),", "))

{a(n) = my(S=x); S = sinh( serreverse( intformal( sqrt(cosh(2*x + O(x^(2*n+2)))) ) )); (2*n-1)!*polcoeff(S,2*n-1)}
for(n=1,20, print1(a(n),", "))

E.g.f.: S(x) = Series_Reversion( Integral sqrt( (1 + 2*x^2) / (1 + x^2) ) dx ).
E.g.f.: S(x) = sinh( Series_Reversion( Integral sqrt( cosh(2*x) ) dx ) ).
Let C(x) be the e.g.f. of A290880, then:
(1) C'(x) = S(x) / sqrt(C(x)^2 + S(x)^2),
(2) S'(x) = C(x) / sqrt(C(x)^2 + S(x)^2),
such that C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1.

A290883 E.g.f. A(x) = sqrt(C(x)^2 + S(x)^2) such that C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1, where C(x) is the e.g.f. of A290880 and S(x) is the e.g.f. of A290881.

Original entry on oeis.org

1, 2, -20, 920, -95600, 17588000, -5034785600, 2068322672000, -1153339941728000, 838147215114560000, -769492266756037760000, 870869784123573927680000, -1191080747725445120960000000, 1936606018449416970940544000000, -3692030834904045806243452160000000, 8156631422332715861303860160000000000, -20671774666617006397027638099614720000000

Offset: 0

Paul D. Hanna, Aug 13 2017

sign

Conjecture: If p is a prime congruent to 1 mod 4 and k is a positive integer, then p^k divides a(n) for n >= (k/2)*p-1. Furthermore, if p^k divides (2n+2)!, then p^k divides a(n). - Andrew Slattery, Aug 21 2022

			E.g.f.: A(x) = 1 + 2*x^2/2! - 20*x^4/4! + 920*x^6/6! - 95600*x^8/8! + 17588000*x^10/10! - 5034785600*x^12/12! + 2068322672000*x^14/14! - 1153339941728000*x^16/16! + 838147215114560000*x^18/18! +...
such that A(x) = sqrt(C(x)^2 + S(x)^2) where series C(x) and S(x) begin:
S(x) = x - x^3/3! + 25*x^5/5! - 1705*x^7/7! + 227665*x^9/9! - 50333425*x^11/11! + 16655398825*x^13/13! - 7711225809625*x^15/15! + 4760499335502625*x^17/17! - 3779764853639958625*x^19/19! + 3752942823715824285625*x^21/21! +...
C(x) = 1 + x^2/2! - 7*x^4/4! + 265*x^6/6! - 24175*x^8/8! + 4037425*x^10/10! - 1070526775*x^12/12! + 412826556025*x^14/14! - 218150106913375*x^16/16! + 151297155973926625*x^18/18! - 133288452772763494375*x^20/20! +...
These series satisfy: C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1.

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A290879, A290880, A290881, A290882, A153302.

{a(n) = my(C=1,S=x); for(i=1,n, C = 1 + intformal( S/sqrt(C^2 + S^2 + O(x^(n+2))) ); S = intformal( C/sqrt(C^2 + S^2)) ); n!*polcoeff(C + S,n)}
for(n=0,30, print1(a(n),", "))

{a(n) = my(E=1); A = sqrt( cosh( 2*serreverse( intformal( sqrt(cosh(2*x + O(x^(2*n+2)))) ) ))); (2*n)!*polcoeff(A,2*n)}
for(n=0,30, print1(a(n),", "))

E.g.f.: A(x) = sqrt( cosh( 2*Series_Reversion( Integral sqrt( cosh(2*x) ) dx ) ) ).

A292181 E.g.f. A(x) satisfies: A(x)^2 + B(x)^2 = C(x)^2, such that A'(x) = A(x) + B(x)*C(x).

Original entry on oeis.org

1, 3, 10, 45, 259, 1806, 14827, 140367, 1504576, 17972559, 236275711, 3387012720, 52572376669, 878552787927, 15729439074058, 300400031036745, 6095885898471775, 130982551821899862, 2970844882925223487, 70929401617621416243, 1778125633605205346584, 46698342082602696345555, 1282168260097348871508667, 36734284970419645262875200, 1096293296048734274708523433, 34026339905854090378353208155

Offset: 1

Paul D. Hanna, Sep 10 2017

nonn

Here, the functions A(x), B(x), and C(x) are the e.g.f.s of sequences A292181, A292182, and A292183, respectively.

Another Pythagorean triple is the e.g.f.s of A289695, A193543, and A153302, which are related to the Lemniscate sine and cosine functions, sl(x) and cl(x).

			E.g.f.: A(x) = x + 3*x^2/2! + 10*x^3/3! + 45*x^4/4! + 259*x^5/5! + 1806*x^6/6! + 14827*x^7/7! + 140367*x^8/8! + 1504576*x^9/9! + 17972559*x^10/10! + 236275711*x^11/11! + 3387012720*x^12/12! + 52572376669*x^13/13! + 878552787927*x^14/14! + 15729439074058*x^15/15! + 300400031036745*x^16/16! +...
where A(x) = Integral A(x) + B(x)*C(x) dx.
RELATED SERIES.
B(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 35*x^4/4! + 226*x^5/5! + 1715*x^6/6! + 14701*x^7/7! + 141248*x^8/8! + 1515661*x^9/9! + 18048527*x^10/10! + 236581984*x^11/11! + 3386091821*x^12/12! + 52533799501*x^13/13! + 877993866290*x^14/14! + 15723411375931*x^15/15! + 300349139257727*x^16/16 +...
where B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
C(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 63*x^4/4! + 361*x^5/5! + 2499*x^6/6! + 20581*x^7/7! + 196311*x^8/8! + 2116561*x^9/9! + 25357563*x^10/10! + 333765037*x^11/11! + 4787007855*x^12/12! + 74323701817*x^13/13! + 1242253733619*x^14/14! + 22243082373301*x^15/15! + 424815246293319*x^16/16! +...
where C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
Squares of series.
A(x)^2 = 2*x^2/2! + 18*x^3/3! + 134*x^4/4! + 1050*x^5/5! + 9158*x^6/6! + 89418*x^7/7! + 972470*x^8/8! + 11700378*x^9/9! + 154613222*x^10/10! + 2227684074*x^11/11! + 34757852054*x^12/12! + 583740365754*x^13/13! + 10497898450118*x^14/14! + 201267889853706*x^15/15! + 4097952119101814*x^16/16! +...
where A(x)^2 + B(x)^2 = C(x)^2.
B(x)^2 = 1 + 2*x + 6*x^2/2! + 26*x^3/3! + 150*x^4/4! + 1082*x^5/5! + 9222*x^6/6! + 89546*x^7/7! + 972726*x^8/8! + 11700890*x^9/9! + 154614246*x^10/10! + 2227686122*x^11/11! + 34757856150*x^12/12! + 583740373946*x^13/13! + 10497898466502*x^14/14! + 201267889886474*x^15/15! + 4097952119167350*x^16/16! +...
where B(x)^2 - A(x)^2 = exp(2*x).
C(x)^2 = 1 + 2*x + 8*x^2/2! + 44*x^3/3! + 284*x^4/4! + 2132*x^5/5! + 18380*x^6/6! + 178964*x^7/7! + 1945196*x^8/8! + 23401268*x^9/9! + 309227468*x^10/10! + 4455370196*x^11/11! + 69515708204*x^12/12! + 1167480739700*x^13/13! + 20995796916620*x^14/14! + 402535779740180*x^15/15! + 8195904238269164*x^16/16! +...
where C(x)^2 - 2*A(x)^2 = exp(2*x).

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A292182 (B), A292183 (C).

{a(n) = my(A=x,B=1,C=1); for(i=0,n, A = intformal(A + B*C + x*O(x^n));
B = 1 + intformal(B + A*C); C = 1 + intformal(C + 2*A*B)); n!*polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

E.g.f. A(x) and related functions B(x) and C(x) satisfy:
(1a) A(x)^2 + B(x)^2 = C(x)^2.
(1b) B(x)^2 - A(x)^2 = exp(x)^2.
(1c) C(x)^2 - 2*A(x)^2 = exp(x)^2.
(2a) A(x) = Integral A(x) + B(x)*C(x) dx.
(2b) B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
(2c) C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
(3a) A(x) = exp(x) * sinh( Integral C(x) dx ).
(3b) B(x) = exp(x) * cosh( Integral C(x) dx ).
(3c) C(x) = exp(x) * cosh( Integral sqrt(2)*B(x) dx).
(3d) A(x) = exp(x) * sinh( Integral sqrt(2)*B(x) dx) / sqrt(2).
(4a) A(x) + B(x) = exp(x) * exp( Integral C(x) dx ).
(4b) C(x) + sqrt(2)*A(x) = exp(x) * exp( Integral sqrt(2)*B(x) dx ).
(4c) C(x) + sqrt(2)*B(x) = (1 + sqrt(2)) * exp(x) * exp( Integral sqrt(2)*A(x) dx ).
(5a) B(x) + i*A(x) = C(x) * exp( i*atan( A(x)/B(x) ) ).
(5b) A(x)/B(x) = Series_Reversion( Integral 1/( sqrt(1-x^4) * (1 + Integral 1/sqrt(1-x^4) dx) ) dx ).
Limit A292182(n)/A292181(n) = 1.
Limit A292183(n)/A292181(n) = sqrt(2).

A292182 E.g.f. B(x) satisfies: A(x)^2 + B(x)^2 = C(x)^2, such that B'(x) = B(x) + A(x)*C(x).

Original entry on oeis.org

1, 1, 2, 7, 35, 226, 1715, 14701, 141248, 1515661, 18048527, 236581984, 3386091821, 52533799501, 877993866290, 15723411375931, 300349139257727, 6095613429234730, 130983518612114231, 2970900143887175977, 70930381205350706888, 1778137090832694851161, 46698407537794612100459, 1282167191852237842607584, 36734238381564939631425737, 1096292258727541156091352361, 34026322932421876848090674594

Offset: 0

Paul D. Hanna, Sep 10 2017

nonn

Here, the functions A(x), B(x), and C(x) are the e.g.f.s of sequences A292181, A292182, and A292183, respectively.

Another Pythagorean triple is the e.g.f.s of A289695, A193543, and A153302, which are related to the Lemniscate sine and cosine functions, sl(x) and cl(x).

			E.g.f.: B(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 35*x^4/4! + 226*x^5/5! + 1715*x^6/6! + 14701*x^7/7! + 141248*x^8/8! + 1515661*x^9/9! + 18048527*x^10/10! + 236581984*x^11/11! + 3386091821*x^12/12! + 52533799501*x^13/13! + 877993866290*x^14/14! + 15723411375931*x^15/15! + 300349139257727*x^16/16 +...
where B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
RELATED SERIES.
A(x) = x + 3*x^2/2! + 10*x^3/3! + 45*x^4/4! + 259*x^5/5! + 1806*x^6/6! + 14827*x^7/7! + 140367*x^8/8! + 1504576*x^9/9! + 17972559*x^10/10! + 236275711*x^11/11! + 3387012720*x^12/12! + 52572376669*x^13/13! + 878552787927*x^14/14! + 15729439074058*x^15/15! + 300400031036745*x^16/16! +...
where A(x) = Integral A(x) + B(x)*C(x) dx.
C(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 63*x^4/4! + 361*x^5/5! + 2499*x^6/6! + 20581*x^7/7! + 196311*x^8/8! + 2116561*x^9/9! + 25357563*x^10/10! + 333765037*x^11/11! + 4787007855*x^12/12! + 74323701817*x^13/13! + 1242253733619*x^14/14! + 22243082373301*x^15/15! + 424815246293319*x^16/16! +...
where C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
Squares of series.
A(x)^2 = 2*x^2/2! + 18*x^3/3! + 134*x^4/4! + 1050*x^5/5! + 9158*x^6/6! + 89418*x^7/7! + 972470*x^8/8! + 11700378*x^9/9! + 154613222*x^10/10! + 2227684074*x^11/11! + 34757852054*x^12/12! + 583740365754*x^13/13! + 10497898450118*x^14/14! + 201267889853706*x^15/15! + 4097952119101814*x^16/16! +...
where A(x)^2 + B(x)^2 = C(x)^2.
B(x)^2 = 1 + 2*x + 6*x^2/2! + 26*x^3/3! + 150*x^4/4! + 1082*x^5/5! + 9222*x^6/6! + 89546*x^7/7! + 972726*x^8/8! + 11700890*x^9/9! + 154614246*x^10/10! + 2227686122*x^11/11! + 34757856150*x^12/12! + 583740373946*x^13/13! + 10497898466502*x^14/14! + 201267889886474*x^15/15! + 4097952119167350*x^16/16! +...
where B(x)^2 - A(x)^2 = exp(2*x).
C(x)^2 = 1 + 2*x + 8*x^2/2! + 44*x^3/3! + 284*x^4/4! + 2132*x^5/5! + 18380*x^6/6! + 178964*x^7/7! + 1945196*x^8/8! + 23401268*x^9/9! + 309227468*x^10/10! + 4455370196*x^11/11! + 69515708204*x^12/12! + 1167480739700*x^13/13! + 20995796916620*x^14/14! + 402535779740180*x^15/15! + 8195904238269164*x^16/16! +...
where C(x)^2 - 2*A(x)^2 = exp(2*x).

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A292181 (A), A292183 (C).

{a(n) = my(A=x,B=1,C=1); for(i=0,n, A = intformal(A + B*C + x*O(x^n));
B = 1 + intformal(B + A*C); C = 1 + intformal(C + 2*A*B)); n!*polcoeff(B,n)}
for(n=0,30,print1(a(n),", "))

E.g.f. B(x) and related functions A(x) and C(x) satisfy:
(1a) A(x)^2 + B(x)^2 = C(x)^2.
(1b) B(x)^2 - A(x)^2 = exp(x)^2.
(1c) C(x)^2 - 2*A(x)^2 = exp(x)^2.
(2a) A(x) = Integral A(x) + B(x)*C(x) dx.
(2b) B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
(2c) C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
(3a) A(x) = exp(x) * sinh( Integral C(x) dx ).
(3b) B(x) = exp(x) * cosh( Integral C(x) dx ).
(3c) C(x) = exp(x) * cosh( Integral sqrt(2)*B(x) dx).
(3d) A(x) = exp(x) * sinh( Integral sqrt(2)*B(x) dx) / sqrt(2).
(4a) A(x) + B(x) = exp(x) * exp( Integral C(x) dx ).
(4b) C(x) + sqrt(2)*A(x) = exp(x) * exp( Integral sqrt(2)*B(x) dx ).
(4c) C(x) + sqrt(2)*B(x) = (1 + sqrt(2)) * exp(x) * exp( Integral sqrt(2)*A(x) dx ).
Limit A292181(n)/A292182(n) = 1.
Limit A292183(n)/A292182(n) = sqrt(2).

A292183 E.g.f. C(x) satisfies: A(x)^2 + B(x)^2 = C(x)^2, such that C'(x) = C(x) + 2A(x)B(x).

Original entry on oeis.org

1, 1, 3, 13, 63, 361, 2499, 20581, 196311, 2116561, 25357563, 333765037, 4787007855, 74323701817, 1242253733619, 22243082373301, 424815246293319, 8620744969300321, 185235767397027627, 4201390722798810493, 100309092062158564959, 2514646421630798317897, 66041388198395188082595, 1813259146315114344920581, 51950114633383773360554679, 1550392693763071812557794801, 48120508780248064233484223067

Offset: 0

Paul D. Hanna, Sep 10 2017

nonn

Here, the functions A(x), B(x), and C(x) are the e.g.f.s of sequences A292181, A292182, and A292183, respectively.

Another Pythagorean triple is the e.g.f.s of A289695, A193543, and A153302, which are related to the Lemniscate sine and cosine functions, sl(x) and cl(x).

			E.g.f.: C(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 63*x^4/4! + 361*x^5/5! + 2499*x^6/6! + 20581*x^7/7! + 196311*x^8/8! + 2116561*x^9/9! + 25357563*x^10/10! + 333765037*x^11/11! + 4787007855*x^12/12! + 74323701817*x^13/13! + 1242253733619*x^14/14! + 22243082373301*x^15/15! + 424815246293319*x^16/16! +...
where C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
RELATED SERIES.
A(x) = x + 3*x^2/2! + 10*x^3/3! + 45*x^4/4! + 259*x^5/5! + 1806*x^6/6! + 14827*x^7/7! + 140367*x^8/8! + 1504576*x^9/9! + 17972559*x^10/10! + 236275711*x^11/11! + 3387012720*x^12/12! + 52572376669*x^13/13! + 878552787927*x^14/14! + 15729439074058*x^15/15! + 300400031036745*x^16/16! +...
where A(x) = Integral A(x) + B(x)*C(x) dx.
B(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 35*x^4/4! + 226*x^5/5! + 1715*x^6/6! + 14701*x^7/7! + 141248*x^8/8! + 1515661*x^9/9! + 18048527*x^10/10! + 236581984*x^11/11! + 3386091821*x^12/12! + 52533799501*x^13/13! + 877993866290*x^14/14! + 15723411375931*x^15/15! + 300349139257727*x^16/16 +...
where B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
Squares of series.
A(x)^2 = 2*x^2/2! + 18*x^3/3! + 134*x^4/4! + 1050*x^5/5! + 9158*x^6/6! + 89418*x^7/7! + 972470*x^8/8! + 11700378*x^9/9! + 154613222*x^10/10! + 2227684074*x^11/11! + 34757852054*x^12/12! + 583740365754*x^13/13! + 10497898450118*x^14/14! + 201267889853706*x^15/15! + 4097952119101814*x^16/16! +...
where A(x)^2 + B(x)^2 = C(x)^2.
B(x)^2 = 1 + 2*x + 6*x^2/2! + 26*x^3/3! + 150*x^4/4! + 1082*x^5/5! + 9222*x^6/6! + 89546*x^7/7! + 972726*x^8/8! + 11700890*x^9/9! + 154614246*x^10/10! + 2227686122*x^11/11! + 34757856150*x^12/12! + 583740373946*x^13/13! + 10497898466502*x^14/14! + 201267889886474*x^15/15! + 4097952119167350*x^16/16! +...
where B(x)^2 - A(x)^2 = exp(2*x).
C(x)^2 = 1 + 2*x + 8*x^2/2! + 44*x^3/3! + 284*x^4/4! + 2132*x^5/5! + 18380*x^6/6! + 178964*x^7/7! + 1945196*x^8/8! + 23401268*x^9/9! + 309227468*x^10/10! + 4455370196*x^11/11! + 69515708204*x^12/12! + 1167480739700*x^13/13! + 20995796916620*x^14/14! + 402535779740180*x^15/15! + 8195904238269164*x^16/16! +...
where C(x)^2 - 2*A(x)^2 = exp(2*x).

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A292181 (A), A292182 (B).

{a(n) = my(A=x,B=1,C=1); for(i=0,n, A = intformal(A + B*C + x*O(x^n));
B = 1 + intformal(B + A*C); C = 1 + intformal(C + 2*A*B)); n!*polcoeff(C,n)}
for(n=0,30,print1(a(n),", "))

E.g.f. C(x) and related functions A(x) and B(x) satisfy:
(1a) A(x)^2 + B(x)^2 = C(x)^2.
(1b) B(x)^2 - A(x)^2 = exp(x)^2.
(1c) C(x)^2 - 2*A(x)^2 = exp(x)^2.
(2a) A(x) = Integral A(x) + B(x)*C(x) dx.
(2b) B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
(2c) C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
(3a) A(x) = exp(x) * sinh( Integral C(x) dx ).
(3b) B(x) = exp(x) * cosh( Integral C(x) dx ).
(3c) C(x) = exp(x) * cosh( Integral sqrt(2)*B(x) dx).
(3d) A(x) = exp(x) * sinh( Integral sqrt(2)*B(x) dx) / sqrt(2).
(4a) A(x) + B(x) = exp(x) * exp( Integral C(x) dx ).
(4b) C(x) + sqrt(2)*A(x) = exp(x) * exp( Integral sqrt(2)*B(x) dx ).
(4c) C(x) + sqrt(2)*B(x) = (1 + sqrt(2)) * exp(x) * exp( Integral sqrt(2)*A(x) dx ).
Limit A292183(n)/A292181(n) = sqrt(2).
Limit A292183(n)/A292182(n) = sqrt(2).

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

Paul D. Hanna, Jan 03 2009

nonn

			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 .

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

{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))}

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

cp's OEIS Frontend

A153300 Coefficient of x^(4n)/(4n)! in the Maclaurin expansion of cm4(x), which is a generalization of the Dixon elliptic function cm(x,0) defined by A104134.

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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.

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A290879 E.g.f. L(x) of aerated sequence satisfies: L(x) = Integral 1 / sqrt( cosh(2*L(x)) ) dx.

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A290880 E.g.f. C(x) satisfies: C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1, where S(x) is the e.g.f. of A290881.

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A290881 E.g.f. S(x) satisfies: C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1, where C(x) is the e.g.f. of A290880.

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A290883 E.g.f. A(x) = sqrt(C(x)^2 + S(x)^2) such that C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1, where C(x) is the e.g.f. of A290880 and S(x) is the e.g.f. of A290881.

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A292181 E.g.f. A(x) satisfies: A(x)^2 + B(x)^2 = C(x)^2, such that A'(x) = A(x) + B(x)*C(x).

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A292183 E.g.f. C(x) satisfies: A(x)^2 + B(x)^2 = C(x)^2, such that C'(x) = C(x) + 2A(x)B(x).