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A278752 E.g.f. D(x) = 1 + Integral S(x)*C(x) dx, where C(x)^2 - S(x)^2 = 1 and 3*C(x)^2 - 2*D(x)^3 = 1. 4
1, 1, 6, 114, 4224, 258696, 23685696, 3030422544, 516368179584, 113039478326016, 30915842271630336, 10330366155858849024, 4141017299122378758144, 1961342355370645525671936, 1083606291089708175858917376, 690681085734140756895484053504, 503068200949361929673857570504704, 415234815803178624028164344747360256, 385549194671700625768876635402899030016 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

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

FORMULA

E.g.f. D(x) and related series S(x) and C(x) satisfy:

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

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

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

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

(5) 3*C(x)^2 - 2*D(x)^3 = 1,

(6) 2*D(x)^3 - 3*S(x)^2 = 2,

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

EXAMPLE

E.g.f.: 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! +...

and related series

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! +...

satisfy

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

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

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)^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! +...

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

PROG

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

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

CROSSREFS

Cf. A278750 (S(x)), A278751 (C(x)), A278749 (C(x) + S(x)).

Sequence in context: A121544 A274786 A317172 * A003425 A052465 A229582

Adjacent sequences:  A278749 A278750 A278751 * A278753 A278754 A278755

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 27 2016

STATUS

approved

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Last modified October 19 12:10 EDT 2019. Contains 328219 sequences. (Running on oeis4.)