A318635 E.g.f. A(x) satisfies: A(A(x)) = sinh( arcsin(x) ).

1, 1, 5, 85, 2825, 151625, 12098125, 1339476125, 196410020625, 37062144900625, 8772471210303125, 2519410212081953125, 854580849916226265625, 348321081954065679265625, 175848355607462110288828125, 95409938858134878259950078125, 39148051664861004366855145390625, 29871598369545793078681814025390625, 108819532565544659211041992639176953125

Offset

1,3

Comments

a(n) modulo 8 has period 4: [1, 1, 5, 5].

a(n) = 5 (mod 10) for n > 2.

First negative term is a(21).

Links

Paul D. Hanna, Table of n, a(n) for n = 1..100

Example

E.g.f.: A(x) = x + x^3/3! + 5*x^5/5! + 85*x^7/7! + 2825*x^9/9! + 151625*x^11/11! + 12098125*x^13/13! + 1339476125*x^15/15! + 196410020625*x^17/17! + 37062144900625*x^19/19! + 8772471210303125*x^21/21! + 2519410212081953125*x^23/23! + 854580849916226265625*x^25/25! +...
such that
A(A(x)) = x + 2*x^3 + 20*x^5 + 520*x^7 + 26000*x^9 + 2132000*x^11 + 260104000*x^13 + 44217680000*x^15 + ... + [Product_{k=1..n-1} ((2*k-1)^2 + 1)] * x^(2*n-1)/(2*n-1)! + ...
which equals sinh( arcsin(x) ).
RELATED SERIES.
Given e.g.f. F(x) of A280790 such that sin( F( sinh( F(x) ) ) ) = x,
then A(x) = sinh( F(x) ) where
F(x) = x + 4*x^5/5! + 2320*x^9/9! + 9857600*x^13/13! + 159122080000*x^17/17! + 7098806416000000*x^21/21! + 686863244097538560000*x^25/25! + ... + A280790(n)*x^(4*n-3)/(4*n-3)! + ...
also sin( F( A(x) ) ) = x.

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, [0, 0]); A[#A] = polcoeff(sinh(asin(x +x*O(x^#A))) - subst(x*Ser(A), x, x*Ser(A)), #A)/2 ); (2*n-1)!*A[2*n-1]}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A280790, A101927.

Sequence in context: A241330 A301435 A280575 * A203800 A277354 A101928

Adjacent sequences: A318632 A318633 A318634 * A318636 A318637 A318638

Keyword

sign

Author

Paul D. Hanna, Sep 05 2018

Status

approved