A258925 E.g.f. satisfies: A(x) = Integral 1 + A(x)^5 dx.

1, 120, 3024000, 858574080000, 1226178516326400000, 5912338932461445120000000, 75732595735526211882516480000000, 2195068320271703663798288449536000000000, 128322069958974226301129597680106864640000000000

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..90

Formula

E.g.f. A(x) satisfies:

(1) A(x) = Series_Reversion( Integral 1/(1+x^5) dx ).

Let C(x) = S'(x) such that S(x) = Series_Reversion( Integral 1/(1-x^5)^(1/5) dx ) is the e.g.f. of A258924, then e.g.f. A(x) of this sequence satisfies:

(2) A(x) = S(x)/C(x),

(3) A(x) = Integral 1/C(x)^5 dx.

From Vaclav Kotesovec, Jun 18 2015: (Start)

a(n) ~ 5^(5*n+5/2) * ((5-sqrt(5))/8)^(5*n/2+5/8) * (5*n)! * n^(1/4) / (sqrt(2) * Gamma(1/4) * Pi^(5*n+5/4)).

a(n) ~ 5^(5*n+5/2) * sin(Pi/5)^(5*n+5/4) * (5*n)! * n^(1/4) / (sqrt(2) * Gamma(1/4) * Pi^(5*n+5/4)).

(End)

Example

E.g.f.: A(x) = x + 120*x^6/6! + 3024000*x^11/11! + 858574080000*x^16/16! +...
where Series_Reversion(A(x)) = x - x^6/6 + x^11/11 - x^16/16 + x^21/21 +...
Also, A(x) = S(x)/C(x) where
S(x) = x - 24*x^6/6! - 169344*x^11/11! - 25255286784*x^16/16! - 23089632627769344*x^21/21! +...+ A258924(n)*x^(5*n+1)/(5*n+1)! +...
C(x) = 1 - 24*x^5/5! - 169344*x^10/10! - 25255286784*x^15/15! - 23089632627769344*x^20/20! +...+ A258924(n)*x^(5*n)/(5*n)! +...
such that C(x)^5 + S(x)^5 = 1.

Prog

(PARI) /* E.g.f. Series_Reversion( Integral 1/(1+x^5) dx ): */
{a(n) = local(A=x); A = serreverse( intformal( 1/(1 + x^5 + O(x^(5*n+2))) ) ); (5*n+1)!*polcoeff(A, 5*n+1)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* E.g.f. A(x) = Integral 1 + A(x)^5 dx.: */
{a(n) = local(A=x); for(i=1, n+1, A = intformal( 1 + A^5 + O(x^(5*n+2)) )); (5*n+1)!*polcoeff(A, 5*n+1)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A258924, A258880, A258901, A258927, A259112, A259113, A258971.

Sequence in context: A172857 A308327 A172897 * A100734 A115463 A003922

Adjacent sequences: A258922 A258923 A258924 * A258926 A258927 A258928

Keyword

nonn

Author

Paul D. Hanna, Jun 15 2015

Status

approved