A104133 -id:A104133 - cp's OEIS frontend

A127974 Numerators in expansion of (1-x)^(-2/3).

1, 2, 5, 40, 110, 308, 2618, 7480, 21505, 559130, 1621477, 4717024, 41273960, 120646960, 353323240, 3109244512, 9133405754, 26862958100, 711868389650, 2098138411600, 6189508314220, 54821359354520, 161972198092900

Offset: 0

Paul Barry, Feb 09 2007

Numerators of n!/A008544(n) are A127975.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A001790, A008544, A104133, A127975.

Numerator[CoefficientList[Series[(1 - x)^(-2/3), {x, 0, 50}], x]] (* G. C. Greubel, May 07 2018 *)

a(n) = denominator(n!/A008544(n)).
a(n) = denominator(n!/(Product_{k=0..n-1} (2+3*k))).
Conjecture: a(n-1) = the numerator of (1/(2*sqrt(3)*Pi)) * Integral_{x >= 0} 1/(1 + x^3)^n. - Peter Bala, Jun 11 2024

A197374 Pi(3): fundamental real period of the Dixonian elliptic functions sm(z) and cm(z).

Original entry on oeis.org

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

Offset: 1

Peter Bala, Mar 04 2012

Pi(3) = 5.29991 62508 56349 87194 ... is the real period of the doubly-periodic Dixonian elliptic functions sm(z) (A104133) and cm(z) (A104134): sm(z+Pi(3)) = sm(z); cm(z+Pi(3)) = cm(z). The other period equals Pi(3)*w, where w = exp(2*I*Pi/3).

			5.2999162508563498719410684989453161077156056146...

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 and P. Flajolet The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion, arXiv:math/0507268v1 [math.CO], 2005; Sem. Lothar. Combin. 54 (2005/06), Art. B54g, 44 pp.
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 330.

Cf. A104133, A104134.

Sqrt[3]/(2*Pi)*Gamma[1/3]^3 // N[#, 103]& // RealDigits // First (* Jean-François Alcover, Jan 21 2013 *)

sqrt(3)/(2*Pi)*gamma(1/3)^3 \\ Charles R Greathouse IV, Mar 04 2012

Pi(3) = 3*int {0..1} 1/(1-t^3)^(2/3) dt = B(1/3,1/3) = Gamma(1/3)^2/Gamma(2/3) = sqrt(3)/(2*Pi)*Gamma(1/3)^3.

Equals Beta(1/3,1/3) (see Shamos). - Stefano Spezia, Jun 03 2025

A381359 E.g.f. A(x) satisfies 1 - A'(x)^2 + 4*A(x)^3 = 0.

Original entry on oeis.org

1, 12, 720, 129600, 51321600, 37977984000, 47113228800000, 90796614543360000, 256892229695692800000, 1021474451008342425600000, 5513370502054734544896000000, 39267642006336798923489280000000, 360478517037545726209161953280000000, 4181620210850033164370074219315200000000

Offset: 0

Paul D. Hanna, Mar 06 2025

nonn

E.g.f. A(x) equals a series trisection of the e.g.f. F(x) of A381360, which satisfies 1 + 3*F'(x)^2 - 4*F(x)^3 = 0. See A381360 for more formulas in which the trisection T1 denotes the e.g.f. A(x) of this sequence.

			E.g.f.: A(x) = x + 12*x^4/4! + 720*x^7/7! + 129600*x^10/10! + 51321600*x^13/13! + 37977984000*x^16/16! + ... + a(n)*x^(3*n+1)/(3*n+1)! + ... where A'(x) = sqrt(1 + 4*A(x)^3).
The ordinary generating function begins
O.g.f.: G(x) = 1 + 12*x + 720*x^2 + 129600*x^3 + 51321600*x^4 + 37977984000*x^5 + 47113228800000*x^6 + ...
Continued fraction representations for the o.g.f. are as follows.
O.g.f.: G(x) = 1/(1 - 12*x/(1 - 48*x/(1 - 150*x/(1 - 294*x/(1 - 576*x/(1 - 900*x/(1 - ...))))))).
O.g.f.: G(x) = 1/(1-12*x - 24^2*x^2/(1-198*x - 210^2*x^2/(1-870*x - 720^2*x^2/(1-2352*x - 1716^2*x^2/(1-4968*x - ...))))).

Paul D. Hanna, Table of n, a(n) for n = 0..200
Roland Bacher and Philippe Flajolet, Pseudo-factorials, elliptic functions, and continued fractions, arXiv:0901.1379 [math.CA], 2009.
Eric van Fossen Conrad and Philippe Flajolet, The Fermat cubic, elliptic functions, continued fractions and a combinatorial excursion, Sem. Lothar. Combin. 54 (2005/06), Art. B54g, 44 pp.

Cf. A381360, A104133, A104134.

{a(n) = my(A = serreverse( intformal( 1/sqrt(1 + 4*x^3 + x*O(x^(3*n+1))) ) )); (3*n+1)!*polcoef(A,3*n+1)}
for(n=0,20, print1(a(n),", "))

E.g.f. A(x) = Sum_{n>=0} a(n)*x^(3*n+1)/(3*n+1)! satisfies the following formulas.
(1) A'(x) = sqrt(1 + 4*A(x)^3).
(2) A'(x) = 1 + 2*Series_Reversion( Integral 1/(1 + x^3)^(2/3) dx )^3.
(3.a) A'(x) = -1 - 2*A(x)/A(-x).
(3.b) A'(x) = -1 + 2*exp( 3*Integral -A(x)*A(-x) dx ).
(4.a) A''(x) = 6*A(x)^2.
(4.b) A'''(x) = 12*A(x)*A'(x).
(4.c) A''''(x) = 12 + 120*A(x)^3.
(5) A(x) = Series_Reversion( Integral 1/sqrt(1 + 4*x^3) dx ).
(6) A(x) = x + 2*Integral Series_Reversion( Integral 1/(1 + x^3)^(2/3) dx )^3 dx.
(7) A(x) = x + Integral Integral 6*A(x)^2 dx dx.
(8.a) A(x) = -x + Integral -2*A(x)/A(-x) dx.
(8.b) A(x) = -A(-x) * exp( 3*Integral -A(x)*A(-x) dx ).
(8.c) A(x) = x * exp( Integral -2/A(-x) - 1/A(x) - 1/x dx ).
(8.d) A(x) = x * exp( Integral sqrt(1 + 4*A(x)^3)/A(x) - 1/x dx ).
(9.a) A(x) = -cm(-x)*sm(-x) where cm(x) and sm(x) are Dixon elliptic functions (A104134 and A104133) satisfying cm(x)^3 + sm(x)^3 = 1.
(9.b) cm(-x) = exp( (1/2)*Integral (A'(x) - 1)/A(x) dx ).
(9.c) sm(-x) = -x*exp( (1/2)*Integral (A'(x) + 1)/A(x) - 2/x dx ).
From Thomas Scheuerle and Paul D. Hanna, Mar 07 2025: (Start)
O.g.f.: 1/T(1, x), where T(k, x) = (1 - 3*k*(3*k-1)^2*x/(1 - 3*k*(3*k+1)^2*x/T(k+1, x))) (continued fraction).
O.g.f.: 1/T(1, x), where T(k, x) = (1 - 3*(2*k - 1)*(9*k^2 - 9*k + 4)*x - ((3*k + 1)*(3*k)*(3*k - 1))^2*x^2/T(k+1, x)) (continued fraction). (End)

A381361 E.g.f. satisfies A(x) = exp( Integral abs(1/A(x)^2) dx ), where abs(F(x)) equals the series expansion formed by the unsigned coefficients in F(x).

Original entry on oeis.org

1, 1, 3, 7, 41, 225, 1435, 11815, 108945, 1062145, 12367475, 154967175, 2052583225, 30729193825, 489419016075, 8162349262375, 149689173742625, 2891193923066625, 58124967786683875, 1262035025301370375, 28655826402568055625, 674057118324247590625, 16913418325411880586875

Offset: 0

Paul D. Hanna, Feb 26 2025

nonn

All terms appear to be odd.

Conjecture: for n > 4, a(n) == 0 (mod 5).

			E.g.f: A(x) = 1 + x/1! + 3*x^2/2! + 7*x^3/3! + 41*x^4/4! + 225*x^5/5! + 1435*x^6/6! + 11815*x^7/7! + 108945*x^8/8! + 1062145*x^9/9! + 12367475*x^10/10! + ...
where A(x) = exp( Integral abs(1/A(x)^2) dx ).
RELATED SERIES.
Compare the signed coefficients of 1/A(x)^2 given by
1/A(x)^2 = 1 - 2*x + 16*x^3/3! - 64*x^4/4! + 2560*x^6/6! - 20480*x^7/7! + 1884160*x^9/9! - 21299200*x^10/10! + 3604480000*x^12/12! + ...
to the unsigned coefficients in log(A(x)) given by
log(A(x)) = x + 2*x^2/2! + 16*x^4/4! + 64*x^5/5! + 2560*x^7/7! + 20480*x^8/8! + 1884160*x^10/10! + 21299200*x^11/11! + 3604480000*x^13/13! + ...
to see that log(A(x)) = Integral abs(1/A(x)^2) dx.
Let B(x) be the e.g.f. of A381360, which starts as
B(x) = 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! + ... + A381360(n)*x^n/n! + ...
then A(x) = B(2*x)^(1/2).
SERIES TRISECTIONS.
The series trisections of A(x) = T0 + T1 + T2 begin
T0 = 1 + 7*x^3/3! + 1435*x^6/6! + 1062145*x^9/9! + 2052583225*x^12/12! + 8162349262375*x^15/15! + 58124967786683875*x^18/18! + ...
T1 = x + 41*x^4/4! + 11815*x^7/7! + 12367475*x^10/10! + 30729193825*x^13/13! + 149689173742625*x^16/16! + 1262035025301370375*x^19/19! + ...
T2 = 3*x^2/2! + 225*x^5/5! + 108945*x^8/8! + 154967175*x^11/11! + 489419016075*x^14/14! + 2891193923066625*x^17/17! + 28655826402568055625*x^20/20! + ...
where (T2^2 + 2*T0*T1)^2 = (T0^2 + 2*T1*T2) * (T1^2 + 2*T0*T2),
also, (T2^2 - T0*T1)^2  = -2 * (T0^2 - T1*T2) * (T1^2 - T0*T2).

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A381360, A104133, A104134.

{a(n) = my(A = sqrt(1 + 2*serreverse( intformal( 1/sqrt(1 + 8*x + 16*x^2 + 32*x^3/3 + x*O(x^n))) ) )); n!*polcoef(A,n)}
for(n=0,25, print1(a(n),", "))

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = sqrt( 1 + 2*Series_Reversion( Integral 1/sqrt(1 + 8*x + 16*x^2 + 32*x^3/3) dx ) ).
(2) A(x) = sqrt( 1 + Integral 2*sqrt( (4*A(x)^6 - 1)/3 ) dx ).
(3) A(x) = sqrt( 1 + 2*x + Integral Integral 8*A(x)^4 dx dx ).
(4) A(x) = exp( x + Integral Integral (4*A(x)^6 + 2)/(3*A(x)^4) dx dx ).
(5) A(x)^4 = (A(x)^2)''/8.
(6.a) 0 = A'(x)^2 + A(x)*A''(x) - 4*A(x)^4.
(6.b) 0 = 1 + 3*A(x)^2*A'(x)^2 - 4*A(x)^6.
(6.c) 0 = 1 - 3*A(x)^3*A''(x) + 8*A(x)^6.
(6.d) 0 = 1 + 2*A(x)^2*A'(x)^2 - A(x)^3*A''(x).
Define the 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) (T2^2 + 2*T0*T1)^2 = (T0^2 + 2*T1*T2) * (T1^2 + 2*T0*T2).
(7.b) (T2^2 - T0*T1)^2  = -2 * (T0^2 - T1*T2) * (T1^2 - T0*T2).
(8) A(x) = 1/sqrt(1 - 2*Integral (T0 - T1) * (A(x) - 3*T2) dx).
(9) A(x) = exp( x + 2*Integral Integral A(x)^2 - (T2^2 + 2*T0*T1) dx dx ).
(10.a) T0^2 + 2*T1*T2 = ((A(x)*A'(x) - 1)/A(x)^2)'/4.
(10.b) T2^2 + 2*T0*T1 = A(x)^2 - (A'(x)/A(x))'/2.
(10.c) T1^2 + 2*T0*T2 = ((A(x)*A'(x) + 1)/A(x)^2)'/4.
(11.a) T0^2 + 2*T1*T2 = A(x)^2/3 + (1 + 3*A(x)*A'(x))/(6*A(x)^4).
(11.b) T2^2 + 2*T0*T1 = A(x)^2/3 - 1/(3*A(x)^4).
(11.c) T1^2 + 2*T0*T2 = A(x)^2/3 + (1 - 3*A(x)*A'(x))/(6*A(x)^4).
(12) A(x) = B(2*x)^(1/2) where B(x) is the e.g.f. of A381360.

A158111 E.g.f.: sm^-1(x) = Sum_{n>=0} a(n)*x^(3n+1)/(3n+1)!; a(n) = coefficient of x^(3n+1)/(3n+1)! in the Maclaurin expansion of the inverse of the Dixon elliptic function sm(x,0).

Original entry on oeis.org

1, 4, 400, 179200, 216832000, 552487936000, 2554704216064000, 19415752042086400000, 225960522265801523200000, 3818732826292045742080000000, 89923520593525093134499840000000, 2854532237720860556461562920960000000, 118891267701073842176624095657984000000000

Offset: 0

Paul D. Hanna, Mar 18 2009

sm(x) = sm(x,0) satisfies: Integral_{y=0..sm(x,0)} dy/(1-y^3)^(2/3) = x.

			E.g.f.: 1/(1-x^3)^(2/3) = 1 + 4*x^3/3! + 400*x^6/6! + 179200*x^9/9! + ...
E.g.f.: sm^-1(x) = x + 4*x^4/4! + 400*x^7/7! + 179200*x^10/10! + ...
sm(x) = x - 4*x^4/4! + 160*x^7/7! - 20800*x^10/10! + 6476800*x^13/13! + ...

Cf. A104133, A004988, A255406.

a(n):= mul(k-0^(mod(k,3)),k=1..3*n):seq(a(n), n = 0 .. 12);
# Peter Bala, Feb 22 2015

Join[{1},Table[Product[(3k-2)(3k-1)^2,{k,n}],{n,14}]] (* Harvey P. Dale, May 19 2012 *)
a[k_] := Pochhammer[2/3, k] (3 k)!/k!; Array[a, 15, 0] (* Jan Mangaldan, Jan 06 2017 *)

PARI
```
a(n)=prod(k=1,n,(3*k-2)*(3*k-1)^2)
```

a(n) = Product_{k=1..n} (3k-2)*(3k-1)^2 for n > 0 with a(0)=1.
E.g.f.: Sum_{n>=0} a(n)*x^(3m)/(3m)! = 1/(1-x^3)^(2/3).
From Peter Bala, Feb 22 2015: (Start)
a(n) = (n - 1/3)! * (3*n)!/( (-1/3)! * n! ).
a(n) = Product {k = 1..3*n} (k - 0^(k mod 3)), where we apply the usual convention that 0^0 = 1. Cf. A255406. (End)
a(n) ~ Gamma(1/3) * 3^(3*n + 1) * n^(3*n + 1/6) / (sqrt(2*Pi) * exp(3*n)). - Vaclav Kotesovec, Apr 10 2018

More terms from Harvey P. Dale, May 19 2012

A261745 Decimal expansion of -sm(-1), where sm(t) is the Dixonian elliptic function sm(t).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Aug 30 2015

In the context of particle physics and in the case of a Yule branching process with two types of particles, this constant appears in the asymptotic expression of the probability that all particles be of the second type at time t as exp(-t)*smh(1).

			1.205415151402983154831411375784488012072704191882249581...

Eric van Fossen Conrad and Philippe Flajolet, The Fermat cubic, elliptic functions, continued fractions and a combinatorial excursion, Séminaire Lotharingien de Combinatoire 54 (2006), Article B54g, page 20.
Eric Weisstein's MathWorld, Weierstrass Elliptic Function

Cf. A104133, A104134.

sm[z_] := 6*WeierstrassP[z, {0, 1/27}]/(1 - 3*WeierstrassPPrime[z, {0, 1/27}]); N[-sm[-1], 105] // RealDigits // First
(* or, without using the Weierstrass P function: *) nint[y_?NumericQ] := NIntegrate[1/(1 + w^3)^(2/3), {w, 0, y}, WorkingPrecision -> 105]; smh[t_] := y /. FindRoot[nint[y] == t, {y, t}, WorkingPrecision -> 105]; N[smh[1], 105] // RealDigits // First

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A127974 Numerators in expansion of (1-x)^(-2/3).

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A197374 Pi(3): fundamental real period of the Dixonian elliptic functions sm(z) and cm(z).

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A381359 E.g.f. A(x) satisfies 1 - A'(x)^2 + 4*A(x)^3 = 0.

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A381361 E.g.f. satisfies A(x) = exp( Integral abs(1/A(x)^2) dx ), where abs(F(x)) equals the series expansion formed by the unsigned coefficients in F(x).

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A158111 E.g.f.: sm^-1(x) = Sum_{n>=0} a(n)*x^(3n+1)/(3n+1)!; a(n) = coefficient of x^(3n+1)/(3n+1)! in the Maclaurin expansion of the inverse of the Dixon elliptic function sm(x,0).

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A261745 Decimal expansion of -sm(-1), where sm(t) is the Dixonian elliptic function sm(t).

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