A063902 -id:A063902 - cp's OEIS frontend

A258895 Decimal expansion of a constant related to A063902 and A258662.

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

Offset: 0

Vaclav Kotesovec, Jun 14 2015

			0.4521042991834205288412808553610327940552545182281851394734731330635...

Cf. A063902, A258662.

evalf(32*Pi / (GAMMA(1/6) * GAMMA(1/3))^2, 118);
evalf(2^(17/3) * Pi^2 / (3 * GAMMA(1/3)^6), 118);

RealDigits[32*Pi/(Gamma[1/6]*Gamma[1/3])^2,10,120][[1]]

Equals limit n->infinity (A063902(n)/(n!)^2)^(1/n).
Equals 32*Pi / (Gamma(1/6) * Gamma(1/3))^2.
Equals 2^(17/3) * Pi^2 / (3 * Gamma(1/3)^6).

A258662 E.g.f. A(x) satisfies: A(x) = exp( Integral A(x) * Integral 1/A(x)^2 dx dx ).

Original entry on oeis.org

1, 1, 4, 40, 760, 23200, 1038400, 64081600, 5214880000, 541085248000, 69718686400000, 10921720817920000, 2044231370959360000, 450550323286412800000, 115495483535461427200000, 34070943029324134912000000, 11460293146666575236608000000, 4360020024970859812710400000000, 1862768688935303816870072320000000

Offset: 0

Paul D. Hanna, Jun 06 2015

nonn

			E.g.f.: A(x) = 1 + x^2/2! + 4*x^4/4! + 40*x^6/6! + 760*x^8/8! + 23200*x^10/10! +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..150

Cf. A063902, A258895.

{a(n) = local(A=1+x); for(i=1, n, A = exp( intformal( A * intformal(1/A^2 + x*O(x^n)) ) ) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(2*n), ", "))

E.g.f.: 1 / [ Sum_{n>=0} (-1)^n * A063902(n) * x^(2*n) / (2*n)! ].
From Vaclav Kotesovec, Jun 14 2015: (Start)
a(n) ~ c * d^n * n!^2 / sqrt(n), where d = 3*A258895 = 96*Pi / (Gamma(1/6) * Gamma(1/3))^2 = 2^(17/3) * Pi^2 / Gamma(1/3)^6 = 1.3563128975502615865238..., c = 0.8047308283353177558313... = 12/(Gamma(1/3)*Gamma(1/6)) = 2^(7/3)*sqrt(3*Pi) / Gamma(1/3)^3.
a(n) ~ 2^(5*n+3) * 3^(n+1) * Pi^(n+1) * n^(2*n+1/2) / (exp(2*n) * Gamma(1/6)^(2*n+1) * Gamma(1/3)^(2*n+1)).
a(n) ~ sqrt(3) * 2^((17*n+10)/3) * Pi^(2*n+3/2) * n^(2*n+1/2) / (exp(2*n) * Gamma(1/3)^(6*n+3)).
(End)

A253649 Coefficients in the expansion of sn(t * x, m) / t in powers of x where t = sqrt( -1/2 - sqrt(1/3)), m = -7 + sqrt(48), and sn() is a Jacobi elliptic function.

Original entry on oeis.org

1, 1, 0, -10, -80, 0, 17600, 418000, 0, -496672000, -23576960000, 0, 91442700800000, 7255463564800000, 0, -69994087116448000000, -8354181454767104000000, 0, 169165728883243642880000000, 28336045031124313753600000000, 0, -1072156342430107319243161600000000

Offset: 0

Michael Somos, May 02 2015

sign

			G.f. = 1 + x - 10*x^3 - 80*x^4 + 17600*x^6 + 418000*x^7 - 496672000*x^9 - ...
E.g.f. = x + x^3/6 - x^7/504 - x^9/4536 + x^13/353808 + 19/59439744*x^15 + ...

Cf. A063902, A144853, A253282.

a[ n_] := If[ n < 0, 0, With[{t = Sqrt[-1/2 - Sqrt[1/3]], m = -7 + Sqrt[48]}, SeriesCoefficient[ JacobiSN[ t x, m] / t, {x, 0, 2 n + 1}] (2 n + 1)! // Simplify]];

{a(n) = my(A, c); if( n<0, 0, A = x + x^3/6; for(k=3, n, A += O(x^(2*k+2)); A = x + intformal( intformal( 2*(A'^2 - 1) / A - A)); c = polcoeff( A, 2*k + 1) * k / (k-2); A = truncate( A + O(x^(2*k))) + c * x^(2*k+1)); (2*n + 1)! * polcoeff( A, 2*n + 1))};

The e.g.f. A(x) = y satisfies 0 = 2 - 2 * y'*y' + y*y'' + y^2.
The e.g.f. A(x) satisfies 0 = A(x) * A(y) * A(x-y) + A(y) * A(z) * A(y-z) - A(x) * A(z) * A(x-z) - A(x-y) * A(x-z) * A(y-z) for all x, y, z.
E.g.f.: Sum_{k>=0} a(k) * x^(2*k+1) / (2*k+1)! = sn(t * x, m) / t where t = sqrt( -1/2 - sqrt(1/3)), m = -7 + sqrt(48), and sn() is a Jacobi elliptic function.
a(3*n + 2) = 0. a(n) = (-1)^floor(n/3) * A063902(n) unless n == 2 (mod 3).

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A258895 Decimal expansion of a constant related to A063902 and A258662.

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A258662 E.g.f. A(x) satisfies: A(x) = exp( Integral A(x) * Integral 1/A(x)^2 dx dx ).

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A253649 Coefficients in the expansion of sn(t * x, m) / t in powers of x where t = sqrt( -1/2 - sqrt(1/3)), m = -7 + sqrt(48), and sn() is a Jacobi elliptic function.

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