A104133 -id:A104133 - cp's OEIS frontend

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.

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 .

A335157 Define b(n) by 1/sm(w) = Sum_{n >= 0} b(3n-1)w^(3n-1), where sm(w) is the elliptic function defined in A104133; a(n) = numerator of b(3n-1).

Original entry on oeis.org

1, 1, -1, -1, 11, 1, -193, -5, 3851, 449, -16493, -17093, 2776513483, 2766847, -18326878991, -284903947, 313476755027, 15306883537, -5759963886461, -549822999679, 43471527926977757, 649802711643571, -53651420037921807347, -278016083032863199, 164833044827776566977996843

Offset: 0

Michel Marcus and N. J. A. Sloane, Jun 08 2020

Note that the expansions of 1/sm(w) and cm(w)/sm(w) on page 4 of the Adams reference agree apart from signs.

			1/sm(w) = w^(-1) + (1/6)*w^2 - (1/252)*w^5 - (1/4536)*w^8 + (11/2476656)*w^11 + (1/3714984)*w^14 - (193/35574686784)*w^17 - (5/15246294336)*w^20 + (3851/582713369521920)*w^23 + (449/1123804355506560)*w^26 - ...

Oscar S. Adams, Elliptic Functions Applied to Conformal World Maps, Special Publication No. 112 of the U.S. Coast and Geodetic Survey, 1925. See pp. 3-4.

Cf. A104133 (sm), A104134 (cm), A335158 (denominators).

A335158 Define b(n) by 1/sm(w) = Sum_{n >= 0} b(3n-1)w^(3n-1), where sm(w) is the elliptic function defined in A104133; a(n) = denominator of b(3n-1).

Original entry on oeis.org

1, 6, 252, 4536, 2476656, 3714984, 35574686784, 15246294336, 582713369521920, 1123804355506560, 2048470579217357568, 35116638500868986880, 283061106066827553906954240, 4665842407694959679784960, 1533625072670071687067878072320, 394360732972304148103168647168

Offset: 0

Michel Marcus and N. J. A. Sloane, Jun 08 2020

			1/sm(w) = w^(-1) + (1/6)*w^2 - (1/252)*w^5 - (1/4536)*w^8 + (11/2476656)*w^11 + (1/3714984)*w^14 - (193/35574686784)*w^17 - (5/15246294336)*w^20 + (3851/582713369521920)*w^23 + (449/1123804355506560)*w^26 - ...

Oscar S. Adams, Elliptic Functions Applied to Conformal World Maps, Special Publication No. 112 of the U.S. Coast and Geodetic Survey, 1925. See pp. 3-4.

Cf. A104133 (sm), A104134 (cm), A335157 (numerators).

A104134 Coefficient of x^(3n)/(3n)! in the Maclaurin expansion of the Dixon elliptic function cm(x,0).

Original entry on oeis.org

1, -2, 40, -3680, 880000, -435776000, 386949376000, -560034421760000, 1233482823823360000, -3926150877331865600000, 17346066637844488192000000, -102987227337891283042304000000, 800183462504065339211776000000000

Offset: 0

Eric van Fossen Conrad (econrad(AT)math.ohio-state.edu), Mar 07 2005

sign

cm(z):=sum((-1)^n*a(n)*z^(3*n)/(3*n)!,n=0..infinity) satisfies sm'(z)=cm(z)^2, cm'(z)=-sm(z)^2 with sm(0)=0 and cm(0)=1. Parametrizes Fermat's cubic X^3+Y^3=1.
Restated with different terminology: the functions sm(x,0) and cm(x,0) satisfy the following initial value problem: d(sm(x,0))/dx = (cm(x,0))^2; d(cm(x,0))/dx = - (sm(x,0))^2; sm(0,0) = 0; cm(0,0) = 1; The functions sm(x,0) and cm(x,0) are elliptic functions which satisfy the equation: (sm(x,0))^3 + (cm(x,0)^3) = 1.
The Dixonian elliptic function cm(z) parametrizes X^3+Y^3=1.

			cm(w) = 1 - (1/3)*w^3 + (1/18)*w^6 - (23/2268)*w^9 + (25/13608)*w^12 - ...

Oscar S. Adams, Elliptic Functions Applied to Conformal World Maps, Special Publication No. 112 of the U.S. Coast and Geodetic Survey, 1925. See p. 3.
A. C. Dixon, On the doubly periodic functions arising out of the curve x^3 + y^3 - 3 alpha xy=1, Quarterly J. Pure Appl. Math. 24 (1890), 167-233.
E. van Fossen Conrad, Some continued fraction expansions of elliptic functions, PhD thesis, The Ohio State University, 2002.

R. Bacher and P. Flajolet, Pseudo-Factorials, Elliptic Functions and Continued Fractions, arXiv:0901.1379 [math.CA], 2009.
P. Flajolet, Publications
E. van Fossen Conrad and P. Flajolet, The Fermat cubic, elliptic functions, continued fractions and a combinatorial excursion, Sem. Lothar. Combin. 54 (2005/06), Art. B54g, 44 pp.
Alessandro Gambini, Giorgio Nicoletti, and Daniele Ritelli, The Wallis Products for Fermat Curves, Vietnam J. Math. (2023).
P. Lindqvist and J. Peetre, Two remarkable identities, called twos, for inverses to some Abelian integrals, Amer. Math. Monthly 108:5, 2001, 403-410.
E. Lundberg, On hypergoniometric functions of complex variables (at Jaak Peetre's home page)

Cf. A104133.

L:=proc(f) expand(x^2*diff(f,y)+y^2*diff(f,x)); end; Lit:=proc(f,m) if m=0 then f else L(Lit(f,m-1)) fi; end; seq(subs(x=0,y=1,Lit(y,3*j)),j=0..20);

nmax = 12; cm[z_] := (3*WeierstrassPPrime[z, {0, 1/27}] + 1) / (3*WeierstrassPPrime[z, {0, 1/27}] - 1); coes = CoefficientList[ Series[ cm[z], {z, 0, 3*nmax}], z][[1 ;; 3*nmax+1]]*Range[0, 3*nmax]!;a[n_] := coes[[3*n+1]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Sep 04 2012 *)
a[ n_] := If[ n < 0, 0, With[ {m = 3 n}, m! SeriesCoefficient[ (3 WeierstrassPPrime[ x, {0, 1/27}] + 1) / (3 WeierstrassPPrime[ x, {0, 1/27}] - 1), {x, 0, m}]] ]; (* Michael Somos, Jun 09 2015 *)
m = 12; is = InverseSeries[Integrate[Normal[1/(1-x^3)^(2/3)+O[x]^(3m)], {x, 0, s}]+O[s]^(3m), s]; a[n_] := Coefficient[(1-is^3)^(1/3), s^(3n)]*(3n)!; a[0] = 1; Table[a[n], {n, 0, m}] (* Jean-François Alcover, Aug 30 2015 *)

{a(n) = my(A, m); if( n<0, 0, A = O(x); for(i=0, n, A = 1 - intformal(intformal(A^2)^2) ); m = 3*n; m! * polcoeff( A, m))}; /* Michael Somos, Aug 17 2007 */

G.f.: cm(u, 0).

E.g.f.: Sum_{k>=0} a(k) * x^(3*k) / (3*k)! = cm(x, 0). - Michael Somos, Aug 17 2007

Additional comments and more terms from Philippe Flajolet, Jul 09 2005
Entry revised by N. J. A. Sloane, Dec 02 2005, Aug 17 2007
Signs added by Michael Somos, Aug 17 2007

A357540 Coefficients T(n,k) of x^(3n+1)r^(3k)/(3n+1)! in power series S(x,r) = Integral C(x,r)^2 * D(x,r)^2 dx such that C(x,r)^3 - S(x,r)^3 = 1 and D(x,r)^3 - r^3*S(x,r)^3 = 1, as a symmetric triangle read by rows.

Original entry on oeis.org

1, 4, 4, 160, 800, 160, 20800, 292800, 292800, 20800, 6476800, 191910400, 500121600, 191910400, 6476800, 3946624000, 210590336000, 1091343616000, 1091343616000, 210590336000, 3946624000, 4161608704000, 361556726784000, 3216369361920000, 6333406238720000, 3216369361920000, 361556726784000, 4161608704000, 6974121256960000, 919365914368000000, 12789764316088320000, 42703786876467200000

Offset: 0

Paul D. Hanna, Oct 09 2022

Related to Dixon elliptic function sm(x,0) (cf. A104133).

			E.g.f.: S(x,r) = Sum_{n>=0} T(n,k) * x^(3*n+1) * r^(3*k) / (3*n+1)! begins:
S(x,r) = Integral C(x,r)^2 * D(x,r)^2 dx = x + (4 + 4*r^3)*x^4/4! + (160 + 800*r^3 + 160*r^6)*x^7/7! + (20800 + 292800*r^3 + 292800*r^6 + 20800*r^9)*x^10/10! + (6476800 + 191910400*r^3 + 500121600*r^6 + 191910400*r^9 + 6476800*r^12)*x^13/13! + (3946624000 + 210590336000*r^3 + 1091343616000*r^6 + 1091343616000*r^9 + 210590336000*r^12 + 3946624000*r^15)*x^16/16! + (4161608704000 + 361556726784000*r^3 + 3216369361920000*r^6 + 6333406238720000*r^9 + 3216369361920000*r^12 + 361556726784000*r^15 + 4161608704000*r^18)*x^19/19! + (6974121256960000 + 919365914368000000*r^3 + 12789764316088320000*r^6 + 42703786876467200000*r^9 + 42703786876467200000*r^12 + 12789764316088320000*r^15 + 919365914368000000*r^18 + 6974121256960000*r^21)*x^22/22! + ...
This table of coefficients T(n,k) of x^(3*n+1) * r^(3*k) / (3*n+1)! in S(x,r) for n >= 0, k = 0..n, begins:
n = 0: [1];
n = 1: [4, 4];
n = 2: [160, 800, 160];
n = 3: [20800, 292800, 292800, 20800];
n = 4: [6476800, 191910400, 500121600, 191910400, 6476800];
n = 5: [3946624000, 210590336000, 1091343616000, 1091343616000, 210590336000, 3946624000];
n = 6: [4161608704000, 361556726784000, 3216369361920000, 6333406238720000, 3216369361920000, 361556726784000, 4161608704000];
n = 7: [6974121256960000, 919365914368000000, 12789764316088320000, 42703786876467200000, 42703786876467200000, 12789764316088320000, 919365914368000000, 6974121256960000];
n = 8: [17455222222028800000, 3313522085749145600000, 67574136526308966400000, 348431220691544883200000, 588750579021316096000000, 348431220691544883200000, 67574136526308966400000, 3313522085749145600000, 17455222222028800000];
...
in which both column 0 and the main diagoal give the unsigned coefficients in the Dixon elliptic function sm(x,0) (cf. A104133).
RELATED SERIES.
C(x,r) = 1 + Integral S(x,r)^2 * D(x,r)^2 dx = 1 + 2*x^3/3! + (40 + 120*r^3)*x^6/6! + (3680 + 37440*r^3 + 21600*r^6)*x^9/9! + (880000 + 20592000*r^3 + 38966400*r^6 + 8553600*r^9)*x^12/12! + (435776000 + 19269888000*r^3 + 79491456000*r^6 + 57708288000*r^9 + 6329664000*r^12)*x^15/15! + (386949376000 + 28748332800000*r^3 + 213892766208000*r^6 + 335872728576000*r^9 + 123646051584000*r^12 + 7852204800000*r^15)*x^18/18! + (560034421760000 + 64544356546560000*r^3 + 774705298498560000*r^6 + 2169194182594560000*r^9 + 1730103155573760000*r^12 + 374841224017920000*r^15 + 15132769090560000*r^18)*x^21/21! + ...
where C(x,r)^3 - S(x,r)^3 = 1.
D(x,r) = 1 + r^3 * Integral S(x,r)^2 * C(x,r)^2 dx = 1 + 2*r^3*x^3/3! + (120*r^3 + 40*r^6)*x^6/6! + (21600*r^3 + 37440*r^6 + 3680*r^9)*x^9/9! + (8553600*r^3 + 38966400*r^6 + 20592000*r^9 + 880000*r^12)*x^12/12! + (6329664000*r^3 + 57708288000*r^6 + 79491456000*r^9 + 19269888000*r^12 + 435776000*r^15)*x^15/15! + (7852204800000*r^3 + 123646051584000*r^6 + 335872728576000*r^9 + 213892766208000*r^12 + 28748332800000*r^15 + 386949376000*r^18)*x^18/18! + (15132769090560000*r^3 + 374841224017920000*r^6 + 1730103155573760000*r^9 + 2169194182594560000*r^12 + 774705298498560000*r^15 + 64544356546560000*r^18 + 560034421760000*r^21)*x^21/21! + ...
where D(x,r)^3 - r^3 * S(x,r)^3 = 1.

Paul D. Hanna, Table of n, a(n) for n = 0..2555

Cf. A104133 (sm(x,0)), A357541 (C(x,r)), A357542 (D(x,r)), A357543 (row sums), A357544 (central terms).

Cf. A357800.

{T(n,k) = my(S=x,C=1,D=1); for(i=0,n,
S = intformal( C^2*D^2 +O(x^(3*n+3)));
C = 1 + intformal( S^2*D^2);
D = 1 + r^3*intformal( S^2*C^2); );
(3*n+1)!*polcoeff( polcoeff(S,3*n+1,x),3*k,r)}
for(n=0,10, for(k=0,n, print1( T(n,k),", "));print(""))

/* Using Series Reversion (faster) */
{T(n,k) = my(S = serreverse( intformal( 1/((1 + x^3)^2*(1 + r^3*x^3)^2 +O(x^(3*n+3)) )^(1/3) )) );
(3*n+1)!*polcoeff( polcoeff(S,3*n+1,x),3*k,r)}
for(n=0,10, for(k=0,n, print1( T(n,k),", "));print(""))

Generating function S(x,r) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(3*n+1) * r^(3*k) / (3*n+1)! and related functions C(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)^3 - S(x,r)^3 = 1.
(1.b) D(x,r)^3 - r^3 * S(x,r)^3 = 1.
(1.c) D(x,r)^3 - r^3 * C(x,r)^3 = 1 - r^3.
Integral formulas.
(2.a) S(x,r) = Integral C(x,r)^2 * D(x,r)^2 dx.
(2.b) C(x,r) = 1 + Integral S(x,r)^2 * D(x,r)^2 dx.
(2.c) D(x,r) = 1 + r^3 * Integral S(x,r)^2 * C(x,r)^2 dx.
(2.d) S(x,r)^3 = Integral 3 * S(x,r)^2 * C(x,r)^2 * D(x,r)^2 dx.
Derivatives.
(3.a) d/dx S(x,r) = C(x,r)^2 * D(x,r)^2.
(3.b) d/dx C(x,r) = S(x,r)^2 * D(x,r)^2.
(3.c) d/dx D(x,r) = r^3 * S(x,r)^2 * C(x,r)^2.
Exponential formulas.
(4.a) C - S = exp( -Integral (C + S) * D^2 dx ).
(4.b) D - r*S = exp( -r * Integral (D + r*S) * C^2 dx ).
(4.c) C + S = sqrt(C^2 - S^2) * exp( Integral D^2/(C^2 - S^2) dx ).
(4.d) D + r*S = sqrt(D^2 - r^2*S^2) * exp( r * Integral C^2/(D^2 - r^2*S^2) dx ).
(5.a) C^2 - S^2 = exp( -2 * Integral S*C/(C + S) * D^2 dx ).
(5.b) D^2 - r^2*S^2 = exp( -2*r^2 * Integral S*D/(D + r*S) * C^2 dx ).
(5.c) C^2 + S^2 = exp( 2 * Integral S*C*(C + S)/(C^2 + S^2) * D^2 dx ).
(5.d) D^2 + r^2*S^2 = exp( 2*r^2 * Integral S*D*(D + r*S)/(D^2 + r^2*S^2) * C^2 dx ).
Hyperbolic functions.
(6.a) C = sqrt(C^2 - S^2) * cosh( Integral D^2/(C^2 - S^2) dx ).
(6.b) S = sqrt(C^2 - S^2) * sinh( Integral D^2/(C^2 - S^2) dx ).
(6.c) D = sqrt(D^2 - r^2*S^2) * cosh( r * Integral C^2/(D^2 - r^2*S^2) dx ).
(6.d) r*S = sqrt(D^2 - r^2*S^2) * sinh( r * Integral C^2/(D^2 - r^2*S^2) dx ).
Other formulas.
(7) S(x,r) = Series_Reversion( Integral ( (1 + x^3)^2 * (1 + r^3*x^3)^2 )^(-1/3) dx ).
(8.a) T(n,0) = T(n,n) = (-1)^n * A104133(n).
(8.b) Sum_{k=0..n} T(n,k) = (3*n+1)!/(3^n*n!) * Product_{k=1..n} (3*k - 2) = A357543(n), for n >= 0.

A098777 Pseudo-factorials: a(0)=1, a(n+1) = (-1)^(n+1) * Sum_{k=0..n} binomial(n,k) * a(k)*a(n-k), n>=0.

Original entry on oeis.org

1, -1, -2, 2, 16, -40, -320, 1040, 12160, -52480, -742400, 3872000, 66457600, -411136000, -8202444800, 58479872000, 1335009280000, -10791497728000, -277035646976000, 2502527565824000, 71391934873600000, -712816377856000000, -22367684235100160000, 244597236078018560000

Offset: 0

Roland Bacher, Oct 04 2004

sign

A variation on the usual factorials (which satisfy the recursion (n+1)!=sum('binomial(n,k)*k!*(n-k)!','k'=0..n) for n>=0).
This sequence seems to satisfy an analog of Wilson's Theorem (which states that (p-1)! equals -1 modulo p for p a prime): For p<10000 a prime congruent to 2 modulo 3, we have a(p-1) congruent to 1 mod p and a(n) congruent to 0 mod p for n>p. For p<10000 a prime congruent to 1 mod 3 we have a(p-1)+a(p) congruent to -1 modulo p.
On the analytic side, the sequence is closely related (via its exponential generating series) to the elliptic curve of j-invariant O (corresponding to the regular hexagonal lattice).
This sequence has a generating function expressible in terms of the Dixon elliptic function sm(x,0) whose coefficients are A104133. The ordinary generating function has a continued fraction expansion of Jacobi type: the numerators are -j^2*(2-(-1)^j) and the denominators are (-1)^(j-1)(j+1/2+(-1)^j/2). - Philippe Flajolet and Roland Bacher, Jan 18 2009

			G.f. = 1 - x - 2*x^2 + 2*x^3 + 16*x^4 - 40*x^5 - 320*x^6 + 1040*x^7 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..464 (first 101 terms from T. D. Noe)
R. Bacher and P. Flajolet, Pseudo-factorials, Elliptic Functions and Continued Fractions, arXiv:0901.1379 [math.CA], 2009.
P. Flajolet, Research Papers
Maxim V. Polyakov, Kirill M. Semenov-Tian-Shansky, Alexander O. Smirnov, Alexey A. Vladimirov, Quasi-Renormalizable Quantum Field Theories, arXiv:1811.08449 [hep-th], 2018.

Cf. A144689, A144750, A235166.

a:= proc(n) option remember; `if`(n=0, 1, (-1)^n*add(
       binomial(n-1, k) *a(k) *a(n-1-k), k=0..n-1))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 22 2018

max = 23; f[z_] = Sum[a[n]*(z^n/n!), {n, 0, max}]; a[0] = 1; a[1] = -1; eq = Rest[ Thread[ CoefficientList[f'[z] + f[-z]^2, z] == 0]]; sol = Solve[ Drop[eq, -max-1]][[1]]; Table[a[n], {n, 0, max}] /. sol (* Jean-François Alcover, Oct 05 2011 *)
a[0] = 1; a[n_] := a[n] = (-1)^n*Sum[Binomial[n-1, k]*a[k]*a[n-k-1], {k, 0, n-1}]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 15 2015 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (1 - 3 InverseSeries[ Integrate[ Series[ (1 - 9 x^2)^(-2/3), {x, 0, n}], x]])^(1/3), {x, 0, n}]]; (* Michael Somos, May 22 2018 *)
Table[SeriesCoefficient[With[{wp = WeierstrassP[z, {0, 4/27}], pd = WeierstrassPPrime[z, {0, 4/27}]}, (2 (2 - 9 pd + 9 wp (2 + 3 pd + 3 wp^2)))/((9 pd + 2 (1 - 3 wp)^2) (2 + 3 wp))], {z, 0, n}] n!, {n, 0, 23}] (* Jan Mangaldan, Jul 07 2020 *)
nmax = 25; CoefficientList[(1 - 3*InverseSeries[Series[x*Hypergeometric2F1[1/2, 2/3, 3/2, 9*x^2], {x, 0, nmax}]])^(1/3), x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 07 2020 *)

a(n)=local(A=1); A=(1-3*serreverse(intformal(1/(1-9*x^2 +x*O(x^n))^(2/3))))^(1/3); n!*polcoeff(A, n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Apr 10 2014

The exponential generating function f(z) = Sum_{n>=0} a(n) * z^n/n! satisfies f'(z)=-f(-z)^2 and is an elliptic function with respect to a regular hexagonal lattice (moreover, -f(z)f(-z) is (up to translation) a Weierstrass function.
a(n) = -n!/R^(n+1)*sum(b^(8*p+4*q)/((p-1/2)*b+(q-1/2)/b)^(n+1), p = -infinity..infinity, q = -infinity..infinity), where b = exp(I*Pi/6) and R = 2^(-4/3)/Pi*GAMMA(1/3)^3. - Philippe Flajolet and Roland Bacher, Jan 18 2009
G.f.: 1/Q(0), where Q(k) = 1 + (2*k+1)*x  +  3*x^2*(2*k+1)^2/(1 - (2*k+1)*x  +  x^2*(2*k+2)^2/Q(k+1) ); (continued fraction after P. Flajolet). - Sergei N. Gladkovskii, Dec 05 2013
E.g.f.: (1 - 3*Series_Reversion( Integral 1/(1 - 9*x^2)^(2/3) dx ))^(1/3). - Paul D. Hanna, Apr 10 2014

A290570 Decimal expansion of Integral_{0..Pi/2} dtheta/(cos(theta)^3 + sin(theta)^3)^(2/3).

Original entry on oeis.org

1, 7, 6, 6, 6, 3, 8, 7, 5, 0, 2, 8, 5, 4, 4, 9, 9, 5, 7, 3, 1, 3, 6, 8, 9, 4, 9, 9, 6, 4, 8, 4, 3, 8, 7, 0, 2, 5, 7, 1, 8, 6, 8, 5, 3, 8, 2, 0, 2, 5, 5, 7, 5, 3, 0, 1, 2, 6, 9, 0, 5, 2, 4, 1, 8, 3, 5, 4, 5, 3, 0, 0, 1, 7, 2, 8, 1, 0, 7, 9, 1, 3, 6, 0, 5, 4, 8, 6, 9, 9, 3, 3, 3, 3, 3, 8, 3, 5, 8, 7, 2, 1, 9, 3, 4

Offset: 1

Jean-François Alcover, Aug 07 2017

			1.766638750285449957313689499648438702571868538202557530126905241835453...

Oscar S. Adams, Elliptic Functions Applied to Conformal World Maps, Special Publication No. 112 of the U.S. Coast and Geodetic Survey, 1925. See constant K p. 9 and previous pages.

StackExchange, Question 2380131 Evaluate in closed form Integral_{0..Pi/2} dtheta/(cos(theta)^3+sin(theta)^3)^(2/3), August 2017.

Cf. A073005 (Gamma(1/3)), A073006 (Gamma(2/3)), A197374 (Beta(1/3,1/3)).

Cf. A104133, A104134, A197374.

RealDigits[(1/3)*Gamma[1/3]^2/Gamma[2/3], 10, 105]

(1/3)*gamma(1/3)^2/gamma(2/3) \\ Michel Marcus, Aug 07 2017

Equals (1/3)*Beta(1/3,1/3).
Equals (1/3)*Gamma(1/3)^2/Gamma(2/3).
Equals A197374/3. - Michel Marcus, Jun 08 2020
From Peter Bala, Mar 01 2022: (Start)
Equals 2*Sum_{n >= 0} (1/(3*n+1) + 1/(3*n-2))*binomial(1/3,n). Cf. A002580 and A175576.
Equals Sum_{n >= 0} (-1)^n*(1/(3*n+1) - 1/(3*n-2))*binomial(1/3,n).
Equals hypergeom([1/3, 2/3], [4/3], 1) = (3/2)*hypergeom([-1/3, -2/3], [4/3], 1) = 2*hypergeom([1/3, 2/3], [4/3], -1) = hypergeom([-1/3, -2/3, 5/6], [4/3, -1/6], -1). (End)

A335180 Define b(n) by 1/cm(w) = Sum_{n >= 0} b(3n)w^(3n), where cm(w) is the elliptic function defined in A104134; a(n) = numerator of b(3n).

Original entry on oeis.org

1, 1, 1, 23, 25, 619, 8083, 584929, 1273037, 202602551, 1543302079, 208247895067, 951782914315, 2366380533924005, 180256368687985157, 12651975031966998901, 55071424621489369589, 3670628209891560101791, 93468279611939477215967, 67638487586433857706623771

Offset: 0

Michel Marcus and N. J. A. Sloane, Jun 10 2020

			1/cm(w) = 1 + (1/3)*w^3 + (1/18)*w^6 + (23/2268)*w^9 + (25/13608)*w^12 + (619/1857492)*w^15 + ...

Oscar S. Adams, Elliptic Functions Applied to Conformal World Maps, Special Publication No. 112 of the U.S. Coast and Geodetic Survey, 1925. See pp. 3-4.

Cf. A104133, A104134, A335157, A335158, A335181.

anum(nn) = my(A = O(x)); for(i=0, nn, A = 1 - intformal(intformal(A^2)^2)); my(v=Vec(1/A)); apply(x->numerator(x), vector(#v\3, k, v[3*k-2])); \\ Michel Marcus, Jun 10 2020

A335181 Define b(n) by 1/cm(w) = Sum_{n >= 0} b(3n)w^(3n), where cm(w) is the elliptic function defined in A104134; a(n) = denominator of b(3n).

Original entry on oeis.org

1, 3, 18, 2268, 13608, 1857492, 133739424, 53362030176, 640344362112, 561902177753280, 23599891465637760, 17558319250434493440, 442469645110949234688, 6065595130003447583720448, 2547549954601447985162588160, 985901832430760370257921617920, 23661643978338248886190118830080

Offset: 0

Michel Marcus and N. J. A. Sloane, Jun 10 2020

			1/cm(w) = 1 + (1/3)*w^3 + (1/18)*w^6 + (23/2268)*w^9 + (25/13608)*w^12 + (619/1857492)*w^15 + ...

Oscar S. Adams, Elliptic Functions Applied to Conformal World Maps, Special Publication No. 112 of the U.S. Coast and Geodetic Survey, 1925. See pp. 3-4.

Cf. A104133, A104134, A335157, A335158, A335180.

aden(nn) = my(A = O(x)); for(i=0, nn, A = 1 - intformal(intformal(A^2)^2)); my(v=Vec(1/A)); apply(x->denominator(x), vector(#v\3, k, v[3*k-2]));

A381360 E.g.f. satisfies A(x) = exp( Integral abs(1/A(x)) dx ), where abs(F(x)) equals the series expansion formed by the unsigned coefficients in F(x).

Original entry on oeis.org

1, 1, 2, 4, 12, 40, 160, 720, 3680, 20800, 129600, 880000, 6476800, 51321600, 435776000, 3946624000, 37977984000, 386949376000, 4161608704000, 47113228800000, 560034421760000, 6974121256960000, 90796614543360000, 1233482823823360000, 17455222222028800000, 256892229695692800000

Offset: 0

Paul D. Hanna, Feb 25 2025

nonn

Series trisections of e.g.f. A(x) are related to Dixon elliptic functions sm(x,0) (A104133) and cm(x,0) (A104134).

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 4*x^3/3! + 12*x^4/4! + 40*x^5/5! + 160*x^6/6! + 720*x^7/7! + 3680*x^8/8! + 20800*x^9/9! + 129600*x^10/10! + 880000*x^11/11! + ...
RELATED SERIES.
Compare the expansion
1/A(x) = 1 - x + 2*x^3/3! - 4*x^4/4! + 40*x^6/6! - 160*x^7/7! + 3680*x^9/9! - 20800*x^10/10! + 880000*x^12/12! -+ ...
to the logarithmic derivative of A(x), which starts as
A'(x)/A(x) = 1 + x + 2*x^3/3! + 4*x^4/4! + 40*x^6/6! + 160*x^7/7! + 3680*x^9/9! + 20800*x^10/10! + 880000*x^12/12! + ...
Compare the expansion of A(x)^2,
A(x)^2 = 1 + 2*x + 6*x^2/2! + 20*x^3/3! + 80*x^4/4! + 360*x^5/5! + 1840*x^6/6! + 10400*x^7/7! + 64800*x^8/8! + ...
to the second derivative A''(x),
A''(x) = 2 + 4*x + 12*x^2/2! + 40*x^3/3! + 160*x^4/4! + 720*x^5/5! + 3680*x^6/6! + 20800*x^7/7! + 129600*x^8/8! + ...
to see that A(x)^2 = A''(x)/2.
The trisections of A(x) = T0(x) + T1(x) + T2(x) begin
T0(x) = 1 + 4*x^3/3! + 160*x^6/6! + 20800*x^9/9! + 6476800*x^12/12! + 3946624000*x^15/15! + ... + (-1)^n*A104133(n)*x^(3*n)/(3*n)! + ...
T1(x) = x + 12*x^4/4! + 720*x^7/7! + 129600*x^10/10! + 51321600*x^13/13! + 37977984000*x^16/16! + ... + A381359(n)*x^(3*n+1)/(3*n+1)! + ...
T2(x) = 2*x^2/2! + 40*x^5/5! + 3680*x^8/8! + 880000*x^11/11! + 435776000*x^14/14! + 386949376000*x^17/17! + ... + (-1)^(n+1)*A104134(n+1)*x^(3*n+2)/(3*n+2)! + ...
where T1(x)^2 = T0(x) * T2(x).
SPECIFIC VALUES.
A(t) = 2 at t = 0.539124944413127749680177459133394238743062994868860...
A(1/2) = 1.88010715382441610819143840438161395486393689452572...
A(1/3) = 1.47706812099920711982922832653100254037067385119264...
A(1/4) = 1.32525993686962199738143806921665823348728083964591...
A(1/5) = 1.24625631796359603424461055708975063363800998581855...
A(1/6) = 1.19796486760762119976757848551541621821975995313722...
A'(1/2) = 2.92023018229509494330357714872624246635412675769217...
  where A'(1/2) = sqrt((4*A(1/2)^3 - 1)/3).
A'(1/3) = 1.99083372061994133511782552227672484323798669077638...
A'(1/4) = 1.66436071755184885803761485601436009074671346271447...
A'(1/5) = 1.49916821151732137928429531008042817934393824335471...
A'(1/6) = 1.39963001300856269549100477471424098446045000735981...
A''(1/2) = 7.0696058197234933075959047886381506980186062076077...
  where A''(1/2) = 2*A(1/2)^2.

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

Cf. A381361 (A(2*x)^(1/2)), A104133 (T0), A104134 (T2), A381359 (T1).

{a(n) = my(A = 1 + serreverse( intformal( 1/sqrt(1 + 4*x + 4*x^2 + 4*x^3/3 + x*O(x^n)) ) )); n!*polcoef(A,n)}
for(n=0,30, print1(a(n),", "))

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + Series_Reversion( Integral 1/sqrt(1 + 4*x + 4*x^2 + 4*x^3/3) dx ).
(2) A(x) = 1 + Integral sqrt( (4*A(x)^3 - 1)/3 ) dx.
(3) A(x) = 1 + x + Integral Integral 2*A(x)^2 dx dx.
(4) A(x) = exp( x + Integral Integral (2*A(x)^3 + 1)/(3*A(x)^2) dx dx ).
(5) A(x)^2 = A''(x)/2.
(6) 0 = 1 + 3*A'(x)^2 - 4*A(x)^3.
Define the series trisections of A(x) = T0 + T1 + T2 by
 T0 = Sum{n>=0} a(3*n)*x^(3*n)/(3*n)!,
 T1 = Sum{n>=0} a(3*n+1)*x^(3*n+1)/(3*n+1)!,
 T2 = Sum{n>=0} a(3*n+2)*x^(3*n+2)/(3*n+2)!,
then these series obey the following formulas.
(7.a) T1^2 = T0 * T2.
(7.b) T0 = (T0^2 - T1*T2)^2.
(7.c) T2 = (T2^2 - T0*T1)^2.
(7.d) T1 = -(T0^2 - T1*T2) * (T2^2 - T0*T1).
(7.e) T0 = (sqrt(T2))' = -(T2^2 - T0*T1)'.
(7.f) T2 = (sqrt(T0))' = (T0^2 - T1*T2)'.
(7.g) T0^(3/2) = 1 + T2^(3/2).
(7.h) T0^3 + T1^3 + T2^3 = 1 + 3*T0*T1*T2.
(7.i) T1' = sqrt(1 + 4*T1^3) = T0^3 - T2^3.
(7.j) T0' = T0 * (sqrt(1 + 4*T1^3) - 1)/T1.
(7.k) T2' = T2 * (sqrt(1 + 4*T1^3) + 1)/T1.
(8.a) T1 = Series_Reversion( Integral 1/(1 + 4*x^3)^(1/2) dx ).
(8.b) T2 = Series_Reversion( Integral 1/(1 + x^3)^(2/3) dx )^2.
(8.c) T0 = d/dx Series_Reversion( Integral 1/(1 + x^3)^(2/3) dx ).
(8.d) T0 = exp( Integral (sqrt(1 + 4*T1^3) - 1)/T1 dx ).
(9.a) A(x) = 1/(1 - Integral T0 - T2 dx).
(9.b) A(x) = exp( x + Integral Integral T0 + T2 dx dx ).
(10.a) T0 = ((A'(x) - 1)/A(x))'/2.
(10.b) T1 = A(x) - (A'(x)/A(x))'.
(10.c) T2 = ((A'(x) + 1)/A(x))'/2.
(11.a) T0 = A(x)/3 + (1 + 3*A'(x))/(6*A(x)^2).
(11.b) T1 = A(x)/3 - 1/(3*A(x)^2).
(11.c) T2 = A(x)/3 + (1 - 3*A'(x))/(6*A(x)^2).
(12.a) T1'' = 6*T1^2.
(12.b) T0'' = 2*(T2^2 + 2*T0*T1).
(12.c) T2'' = 2*(T0^2 + 2*T1*T2).

cp's OEIS Frontend

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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A335157 Define b(n) by 1/sm(w) = Sum_{n >= 0} b(3*n-1)*w^(3*n-1), where sm(w) is the elliptic function defined in A104133; a(n) = numerator of b(3*n-1).

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A335158 Define b(n) by 1/sm(w) = Sum_{n >= 0} b(3*n-1)*w^(3*n-1), where sm(w) is the elliptic function defined in A104133; a(n) = denominator of b(3*n-1).

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A104134 Coefficient of x^(3n)/(3n)! in the Maclaurin expansion of the Dixon elliptic function cm(x,0).

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A357540 Coefficients T(n,k) of x^(3*n+1)*r^(3*k)/(3*n+1)! in power series S(x,r) = Integral C(x,r)^2 * D(x,r)^2 dx such that C(x,r)^3 - S(x,r)^3 = 1 and D(x,r)^3 - r^3*S(x,r)^3 = 1, as a symmetric triangle read by rows.

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A098777 Pseudo-factorials: a(0)=1, a(n+1) = (-1)^(n+1) * Sum_{k=0..n} binomial(n,k) * a(k)*a(n-k), n>=0.

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A290570 Decimal expansion of Integral_{0..Pi/2} dtheta/(cos(theta)^3 + sin(theta)^3)^(2/3).

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A335157 Define b(n) by 1/sm(w) = Sum_{n >= 0} b(3n-1)w^(3n-1), where sm(w) is the elliptic function defined in A104133; a(n) = numerator of b(3n-1).

A335158 Define b(n) by 1/sm(w) = Sum_{n >= 0} b(3n-1)w^(3n-1), where sm(w) is the elliptic function defined in A104133; a(n) = denominator of b(3n-1).

A357540 Coefficients T(n,k) of x^(3n+1)r^(3k)/(3n+1)! in power series S(x,r) = Integral C(x,r)^2 * D(x,r)^2 dx such that C(x,r)^3 - S(x,r)^3 = 1 and D(x,r)^3 - r^3*S(x,r)^3 = 1, as a symmetric triangle read by rows.

A335180 Define b(n) by 1/cm(w) = Sum_{n >= 0} b(3n)w^(3n), where cm(w) is the elliptic function defined in A104134; a(n) = numerator of b(3n).

A335181 Define b(n) by 1/cm(w) = Sum_{n >= 0} b(3n)w^(3n), where cm(w) is the elliptic function defined in A104134; a(n) = denominator of b(3n).