|
| |
|
|
A200030
|
|
G.f. satisfies: A(x) = 1 + x*A(x)^3 - 2*x^2*A(x)^2 + x^3*A(x).
|
|
2
|
|
|
|
1, 1, 1, 3, 11, 42, 167, 685, 2879, 12338, 53718, 236961, 1056825, 4757575, 21590295, 98665559, 453663399, 2097270984, 9742489832, 45452945649, 212884993100, 1000597657272, 4718101921367, 22312506322722, 105802828419669, 502945922930746, 2396283673887013, 11441289644315619
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. satisfies: A(x) = 1 + x*A(x)*(A(x) - x)^2.
Recurrence: 2*n*(2*n+1)*(755*n^4 - 9674*n^3 + 42481*n^2 - 74770*n + 42720)*a(n) = 3*(6795*n^6 - 93861*n^5 + 471409*n^4 - 1076251*n^3 + 1146308*n^2 - 542024*n + 88632)*a(n-1) - 18*(1510*n^6 - 23123*n^5 + 134831*n^4 - 376691*n^3 + 516039*n^2 - 303758*n + 42768)*a(n-2) + 2*(n-4)*(3020*n^5 - 38696*n^4 + 182101*n^3 - 375490*n^2 + 299427*n - 29106)*a(n-3) + 36*(n-5)*(2*n-3)*(7*n^2 - 3*n + 38)*a(n-4) + 4*(n-6)*(755*n^5 - 10429*n^4 + 50757*n^3 - 103199*n^2 + 80998*n - 18342)*a(n-5) - 2*(n-7)*(2*n-11)*(755*n^4 - 6654*n^3 + 17989*n^2 - 15810*n + 1512)*a(n-6). - Vaclav Kotesovec, Sep 11 2013
a(n) ~ c*d^n/n^(3/2), where d = 5.046285698650584886... is the root of the equation 4 - 4*d - 8*d^3 + 36*d^4 - 27*d^5 + 4*d^6 = 0 and c = 0.173559437769138492910993971173612172... - Vaclav Kotesovec, Sep 11 2013
|
|
|
EXAMPLE
|
G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 11*x^4 + 42*x^5 + 167*x^6 + 685*x^7 +...
Related expansions:
A(x)^2 = 1 + 2*x + 3*x^2 + 8*x^3 + 29*x^4 + 112*x^5 + 449*x^6 +...
A(x)^3 = 1 + 3*x + 6*x^2 + 16*x^3 + 57*x^4 + 222*x^5 + 898*x^6 +...
|
|
|
MATHEMATICA
|
nmax=20; aa=ConstantArray[0, nmax]; aa[[1]]=1; Do[AGF=1+Sum[aa[[n]]*x^n, {n, 1, j-1}]+koef*x^j; sol=Solve[Coefficient[1 + x*AGF*(AGF - x)^2 - AGF, x, j]==0, koef][[1]]; aa[[j]]=koef/.sol[[1]], {j, 2, nmax}]; Flatten[{1, aa}] (* Vaclav Kotesovec, Sep 11 2013 *)
|
|
|
PROG
|
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A^3-2*x^2*A^2+x^3*A^1+x*O(x^n)); polcoeff(A, n)}
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
