A205319 - OEIS

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

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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 * A377790 A355099 A209917

Adjacent sequences: A205316 A205317 A205318 * A205320 A205321 A205322

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 25 2012

STATUS

approved