A322895 E.g.f. S(x) = Integral C(x) * C(S(x)) dx, such that C(x)^2 - S(x)^2 = 1, where S(x) = Sum_{n>=0} a(n)*x^(2*n+1)/(2*n+1)!, with coefficients a(n) starting at n = 0.

1, 2, 24, 872, 67072, 9174400, 1999010432, 644045742336, 290850932891648, 176867741048885248, 140377061404214788096, 141829845629449484697600, 178724167497716114197741568, 276141744068786710349406863360, 515617816085923457367906068463616, 1149118315292952171200930347988287488, 3023691286171534277132478231960440799232

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..150

Formula

E.g.f. S(x) and related series C(x) satisfy the following relations.

(1a) S(x) = Integral C(x) * C(S(x)) dx.

(1b) C(x) = 1 + Integral S(x) * C(S(x)) dx.

(2) C(x)^2 - S(x)^2 = 1.

(3a) d/dx S(x) = C(x) * C(S(x)).

(3b) d/dx C(x) = S(x) * C(S(x)).

(4a) C(x) + S(x) = exp( Integral C(S(x)) dx ).

(4b) C(x) = cosh( Integral C(S(x)) dx ).

(4c) S(x) = sinh( Integral C(S(x)) dx ).

(5) C(S(x))^2 - S(S(x))^2 = 1.

(5a) S(S(x)) = Integral C(x) * C(S(x))^2 * C(S(S(x))) dx.

(5b) C(S(x)) = 1 + Integral C(x) * S(S(x)) * C(S(x)) * C(S(S(x))) dx.

(6a) C(S(x)) + S(S(x)) = exp( Integral C(x) * C(S(x)) * C(S(S(x))) dx ).

(6b) C(S(x)) = cosh( Integral C(x) * C(S(x)) * C(S(S(x))) dx ).

(6c) S(S(x)) = sinh( Integral C(x) * C(S(x)) * C(S(S(x))) dx ).

(7) C(S(S(x))) + S(S(S(x))) = exp( Integral C(x) * C(S(x))^2 * C(S(S(x))) * C(S(S(S(x)))) dx ).

Example

E.g.f. S(x) = x + 2*x^3/3! + 24*x^5/5! + 872*x^7/7! + 67072*x^9/9! + 9174400*x^11/11! + 1999010432*x^13/13! + 644045742336*x^15/15! + 290850932891648*x^17/17! + ...
such that S(x) = Integral C(x) * C(S(x)) dx.
RELATED SERIES.
C(x) = 1 + x^2/2! + 5*x^4/4! + 109*x^6/6! + 5737*x^8/8! + 579961*x^10/10! + 98213933*x^12/12! + 25474555941*x^14/14! + 9505761607249*x^16/16! + 4872947687449969*x^18/18! + ...
such that C(x)^2 - S(x)^2 = 1.
C(x) + S(x) = 1 + x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 24*x^5/5! + 109*x^6/6! + 872*x^7/7! + 5737*x^8/8! + 67072*x^9/9! + 579961*x^10/10! + 9174400*x^11/11! + 98213933*x^12/12! + 1999010432*x^13/13! + 25474555941*x^14/14! + 644045742336*x^15/15! + 9505761607249*x^16/16! + 290850932891648*x^17/17! + 4872947687449969*x^18/18! + ...
such that C(x) + S(x) = exp( Integral C(S(x)) dx ).
C(S(x)) = 1 + x^2/2! + 13*x^4/4! + 493*x^6/6! + 39929*x^8/8! + 5724249*x^10/10! + 1299323781*x^12/12! + 433635007877*x^14/14! + 201870080039537*x^16/16! + ...
S(S(x)) = x + 4*x^3/3! + 88*x^5/5! + 4992*x^7/7! + 549504*x^9/9! + 101239168*x^11/11! + 28464335360*x^13/13! + 11465663251456*x^15/15! + 6319308066455552*x^17/17! + ...
C(S(x)) + S(S(x)) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 88*x^5/5! + 493*x^6/6! + 4992*x^7/7! + 39929*x^8/8! + 549504*x^9/9! + 5724249*x^10/10! + 101239168*x^11/11! + 1299323781*x^12/12! + 28464335360*x^13/13! + 433635007877*x^14/14! + 11465663251456*x^15/15! + 201870080039537*x^16/16! + 6319308066455552*x^17/17! + ...
such that C(S(x)) + S(S(x)) = exp( Integral C(x) * C(S(x)) * C(S(S(x))) dx ).
If H(H(x)) = S(x) then
H(x) = x + x^3/3! + 7*x^5/5! + 205*x^7/7! + 13305*x^9/9! + 1616133*x^11/11! + 320304759*x^13/13! + 95177183745*x^15/15! + 40025542374641*x^17/17! + 22825140776633385*x^19/19! + 17079280074768716487*x^21/21! + 16337152342909182929909*x^23/23! + 19558206881883825876978857*x^25/25! + 28793090340440086848693036589*x^27/27! + 51357088945721875208166952420407*x^29/29! + ...
the nonzero coefficients of which appear to consist of only odd numbers.

Prog

(PARI) {a(n) = my(S=x, C=1); for(i=1, 2*n,
S = intformal( C * subst(C, x, S) + x*O(x^(2*n)) );
C = 1 + intformal( S * subst(C, x, S) + x*O(x^(2*n)) ); );
(2*n+1)! * polcoeff( S, 2*n+1)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A322896 (C), A322897 (C+S).

Sequence in context: A265879 A339946 A172492 * A264559 A012186 A012081

Adjacent sequences: A322892 A322893 A322894 * A322896 A322897 A322898

Keyword

nonn

Author

Paul D. Hanna, Jan 06 2019

Status

approved