A278749 E.g.f. C(x) + S(x), such that C(x)^2 - S(x)^2 = 1, 3*C(x)^2 - 2*D(x)^3 = 1, and D(x) = 1 + Integral S(x)*C(x) dx.

1, 1, 1, 3, 9, 39, 189, 1137, 7521, 58221, 487521, 4615623, 46747449, 523484019, 6218441469, 80413567317, 1095843999681, 16072230046041, 247107215918241, 4053246141598443, 69395454770712489, 1258826280827924799, 23749475226740969949, 472083799922946212697, 9730211267060692468641, 210327336751547848824261, 4701988645468367963255361, 109812853605044722106919663

Offset

0,4

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

E.g.f. A(x) = C(x) + S(x), where S(x), C(x), and D(x) satisfy:

(1) A(x) = exp( Integral D(x)^2 dx ),

(2) A(x) = 1 + Integral A(x)*D(x)^2 dx,

(3) S(x) = Integral C(x)*D(x)^2 dx,

(4) C(x) = 1 + Integral S(x)*D(x)^2 dx,

(5) D(x) = 1 + Integral S(x)*C(x) dx,

(6) C(x)^2 - S(x)^2 = 1,

(7) 3*C(x)^2 - 2*D(x)^3 = 1,

(8) 2*D(x)^3 - 3*S(x)^2 = 2.

Example

E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 39*x^5/5! + 189*x^6/6! + 1137*x^7/7! + 7521*x^8/8! + 58221*x^9/9! + 487521*x^10/10! + 4615623*x^11/11! + 46747449*x^12/12! +...
such that A(x) = C(x) + S(x) where
S(x) = x + 3*x^3/3! + 39*x^5/5! + 1137*x^7/7! + 58221*x^9/9! + 4615623*x^11/11! + 523484019*x^13/13! + 80413567317*x^15/15! + 16072230046041*x^17/17! + 4053246141598443*x^19/19! +...
C(x) = 1 + x^2/2! + 9*x^4/4! + 189*x^6/6! + 7521*x^8/8! + 487521*x^10/10! + 46747449*x^12/12! + 6218441469*x^14/14! + 1095843999681*x^16/16! + 247107215918241*x^18/18! +...
Related expansions
C(x)^2 = 1 + 2*x^2/2! + 24*x^4/4! + 648*x^6/6! + 31296*x^8/8! + 2366352*x^10/10! + 257865984*x^12/12! + 38266414848*x^14/14! + 7419295374336*x^16/16! + 1820980419409152*x^18/18! +...
D(x) = 1 + x^2/2! + 6*x^4/4! + 114*x^6/6! + 4224*x^8/8! + 258696*x^10/10! + 23685696*x^12/12! + 3030422544*x^14/14! + 516368179584*x^16/16! + 113039478326016*x^18/18! +...
D(x)^2 = 1 + 2*x^2/2! + 18*x^4/4! + 408*x^6/6! + 17352*x^8/8! + 1184832*x^10/10! + 118618128*x^12/12! + 16371203328*x^14/14! + 2979295540992*x^16/16! + 691248148134912*x^18/18! +...
D(x)^3 = 1 + 3*x^2/2! + 36*x^4/4! + 972*x^6/6! + 46944*x^8/8! + 3549528*x^10/10! + 386798976*x^12/12! + 57399622272*x^14/14! + 11128943061504*x^16/16! + 2731470629113728*x^18/18! +...
satisfy
C(x)^2 - S(x)^2 = 1,
3*C(x)^2 - 2*D(x)^3 = 1.
Logarithm of the e.g.f. begins:
log(A(x)) = x + 2*x^3/3! + 18*x^5/5! + 408*x^7/7! + 17352*x^9/9! + 1184832*x^11/11! + 118618128*x^13/13! + 16371203328*x^15/15! +...
which equals Integral D(x)^2 dx.

Prog

(PARI) {a(n) = my(S=x, C=1, D=1); for(i=1, n, S = intformal(C*(D^2 +O(x^(n+1)))); C = 1 + intformal(S*(D^2 +O(x^(n+1)))); D = 1 + intformal(S*C); ); n!*polcoeff(C+S, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A278750 (S(x)), A278751 (C(x)), A278752 (D(x)).

Sequence in context: A360876 A287063 A080635 * A208816 A130905 A030799

Adjacent sequences: A278746 A278747 A278748 * A278750 A278751 A278752

Keyword

nonn

Author

Paul D. Hanna, Nov 27 2016

Status

approved