A205319 G.f.: A(x) = x + x*ITERATE^3(x + x*ITERATE^5(x + x*ITERATE^7(x + x*ITERATE^9(x + ...)))), where ITERATE^n(F(x)) denotes the n-th iteration of F(x), and the nesting of odd iterations continue indefinitely.

1, 1, 3, 21, 249, 4265, 96773, 2759229, 95447863, 3906844257, 185703476509, 10101591706057, 621466309755017, 42823493907501689, 3278179258815893143, 276843784084034075045, 25636587602145118330341, 2589449871609662090384097, 283950074290110976428751741

Offset

1,3

Links

Table of n, a(n) for n=1..19.

Example

G.f.: A(x) = x + x^2 + 3*x^3 + 21*x^4 + 249*x^5 + 4265*x^6 + 96773*x^7 +...
where A(x) is generated by nesting odd iterations of shifted series:
A(x) = x + x*B(B(B(x)));
B(x) = x + x*C(C(C(C(C(x)))));
C(x) = x + x*D(D(D(D(D(D(D(x)))))));
D(x) = x + x*E(E(E(E(E(E(E(E(E(x))))))))); ...
The coefficients in the respective series begin:
B: [1, 1, 5, 55, 945, 22025, 648165, 23096609, 969937489, ...];
C: [1, 1, 7, 105, 2373, 70553, 2580977, 111821185, 5601348627, ...];
D: [1, 1, 9, 171, 4797, 173913, 7637649, 391725549, 22936666197, ...];
E: [1, 1, 11, 253, 8481, 363209, 18648685, 1107719349, 74460444575, ...];
F: [1, 1, 13, 351, 13689, 676585, 39800813, 2690422761, 204504278537, ...];
G: [1, 1, 15, 465, 20685, 1159225, 76905145, 5833316697, 495166782107, ...];
H: [1, 1, 17, 595, 29733, 1863353, 137665337, 11583449749, 1086710817325, ...]; ...

Prog

(PARI) /* Define the n-th iteration of function F: */
{ITERATE(n, F, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}
/* G.f. A(x) results from nested iterations of shifted series: */
{a(n)=local(A=x); for(k=0, n, A=x + x*ITERATE(2*n-2*k+3, A, n)); polcoeff(A, n)}

Crossrefs

Cf. A195192, A205320.

Sequence in context: A365602 A371006 A355092 * A355099 A209917 A179504

Adjacent sequences: A205316 A205317 A205318 * A205320 A205321 A205322

Keyword

nonn

Author

Paul D. Hanna, Jan 25 2012

Status

approved