A322895 - OEIS

%I #12 Jan 09 2019 13:43:15

%S 1,2,24,872,67072,9174400,1999010432,644045742336,290850932891648,

%T 176867741048885248,140377061404214788096,141829845629449484697600,

%U 178724167497716114197741568,276141744068786710349406863360,515617816085923457367906068463616,1149118315292952171200930347988287488,3023691286171534277132478231960440799232

%N 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.

%H Paul D. Hanna, <a href="/A322895/b322895.txt">Table of n, a(n) for n = 0..150</a>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%F (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 ).

%e 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! + ...

%e such that S(x) = Integral C(x) * C(S(x)) dx.

%e RELATED SERIES.

%e 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! + ...

%e such that C(x)^2 - S(x)^2 = 1.

%e 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! + ...

%e such that C(x) + S(x) = exp( Integral C(S(x)) dx ).

%e 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! + ...

%e 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! + ...

%e 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! + ...

%e such that C(S(x)) + S(S(x)) = exp( Integral C(x) * C(S(x)) * C(S(S(x))) dx ).

%e If H(H(x)) = S(x) then

%e 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! + ...

%e the nonzero coefficients of which appear to consist of only odd numbers.

%o (PARI) {a(n) = my(S=x,C=1); for(i=1,2*n,

%o S = intformal( C * subst(C,x,S) + x*O(x^(2*n)) );

%o C = 1 + intformal( S * subst(C,x,S) + x*O(x^(2*n)) ););

%o (2*n+1)! * polcoeff( S, 2*n+1)}

%o for(n=0,20, print1(a(n),", "))

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 06 2019