|
| |
|
|
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
|
|
|
|
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
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
