A281435 E.g.f. S(x) satisfies: S(x) = Integral (1 + S(x)^2)^(7/2) dx.

1, 7, 301, 32347, 6476281, 2080072687, 978357441061, 633671918506627, 540647648053353841, 587611403828850167767, 792504001599034713809821, 1298643416767164198145121707, 2541154725546790383213482500201, 5852376595008692595588309106586047, 15669400182952350735653156519506572181, 48261540873448422135971738449165162450387, 169430327422451431526680101559949211638388961

Offset

1,2

Links

Table of n, a(n) for n=1..17.

Diego Dominici, Nested derivatives: a simple method for computing series expansions of inverse functions, Int. Jour. of Mathematics and Mathematical Sciences, Vol. 2003, Issue 58, pp. 3699-3715. arXiv:math/0501052v2 [math.CA], 2005.

Formula

C(x)^2 - S(x)^2 = 1 and S'(x) = C(x)^7, where C(x) is described by A281436.

From Peter Bala, Dec 22 2025: (Start)

For n >= 1, define f(n, x) = d/dx( (1 + x^2)^(7/2) * f(n-1, x) ) with f(0, x) = 1. Then

a(n) = f(2*n, 0) (apply Dominici, Theorem 4.1, to S(x) = Series_Reversion( Integral 1/(1 + x^2)^(7/2) dx ).

Define T(x) = S(x)/C(x). Then T(x) = Series_Reversion( Integral ( 1 - x^2)^2 dx ).

The e.g.fs. S(x), C(x) and T(x) are algebraic functions of x:

(225*x^2 - 64)*S(x)^10 + 5*(225*x^2 - 64)*S(x)^8 + 10*(225*x^2 - 64)*S(x)^6 + 150*(15*x^2 - 4)*S(x)^4 + 225*(5*x^2 - 1)*S(x)^2 + 225*x^2 = 0 with S(0) = 0.

(225*x^2 - 64)*C(x)^10 + 40*C(x)^4 + 15*C(x)^2 + 9 = 0 with C(0) = 1.

3*T(x)^5 - 10*T(x)^3 + 15*T(x) - 15*x = 0 with T(0) = 0.

15*x*C(x)^5 - 15*S(x)*C(x)^4 + 10*C(x)^2*S(x)^3 - 3*S(x)^5 = 0. (End)

Example

From Peter Bala, Dec 23 2025: (Start)
S(x) = x + 7*x^3/3! + 301*x^5/5! + 32347*x^7/7! + 6476281*x^9/9! + 2080072687*x^11/11! + 978357441061*x^13/13! + 633671918506627*x^15/15! + 540647648053353841*x^17/17! + ...
C(x) = 1 + x^2/2! + 25*x^4/4! + 1921*x^6/6! + 301105*x^8/8! + 79715041*x^10/10! + 31953352585*x^12/12! + 18055521444961*x^14/14! + 13675020394100065*x^16/16! + 13371875150000047681*x^18/18! + ...
T(x) = x + 4*x^3/3! + 136*x^5/5! + 12544*x^7/7! + 2240896*x^9/9! + 657040384*x^11/11! + 286411181056*x^13/13! + 173782339354624*x^15/15! + 140023837703766016*x^17/17! + ....
(End)

Maple

f := proc (n, x) option remember; if n = 0 then 1 else simplify( diff( (1 + x^2)^(7/2) * f(n-1, x), x ) ) end if end proc:
seq( eval(f(2*n, x), x = 0), n = 0..20 ); # Peter Bala, Dec 22 2025

Prog

(PARI) {a(n) = my(S=x, C=1); for(i=1, n, S = intformal( C^7 +x*O(x^(2*n))); C = 1 + intformal( S*C^6 ) ); (2*n-1)!*polcoeff(S, 2*n-1)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A281181, A281182, A281427, A281428, A281436, A281443, A281444.

Sequence in context: A082168 A362658 A096348 * A015005 A209806 A257919

Adjacent sequences: A281432 A281433 A281434 * A281436 A281437 A281438

Keyword

nonn,easy

Author

Paul D. Hanna, Jan 21 2017

Status

approved