A205320 G.f.: A(x) = x + x*ITERATE^2(x + x*ITERATE^4(x + x*ITERATE^6(x + x*ITERATE^8(x + ...)))), where ITERATE^n(F(x)) denotes the n-th iteration of F(x), and the nesting of even iterations continue indefinitely.

1, 1, 2, 10, 93, 1328, 26122, 662688, 20768108, 780250074, 34376313158, 1746482796840, 100964618760718, 6570171769295176, 476961360708388461, 38334059714026619028, 3388777691630610169520, 327633680973521183134333, 34470692910252290655821322, 3929174185110859040857539276

Offset

1,3

Links

Table of n, a(n) for n=1..20.

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 10*x^4 + 93*x^5 + 1328*x^6 + 26122*x^7 +...
where A(x) is generated by nesting even iterations of shifted series:
A(x) = x + x*B(B(x)));
B(x) = x + x*C(C(C(C(x)))));
C(x) = x + x*D(D(D(D(D(D(x)))))));
D(x) = x + x*E(E(E(E(E(E(E(E(x))))))))); ...
The coefficients in the respective series begin:
B: [1, 1, 4, 36, 522, 10528, 273524, 8744724, 333829460, ...];
C: [1, 1, 6, 78, 1551, 41104, 1355350, 53445516, 2457584702, ...];
D: [1, 1, 8, 136, 3444, 113600, 4570568, 216110200, 11732501680, ...];
E: [1, 1, 10, 210, 6465, 255600, 12172130, 673488008, 42339965808, ...];
F: [1, 1, 12, 300, 10878, 501728, 27635132, 1754547372, 125631525316, ...];
G: [1, 1, 14, 406, 16947, 893648, 55924974, 4010847764, 322635273394, ...];
H: [1, 1, 16, 528, 24936, 1480064, 103765520, 8300468272, 741606109248, ...]; ...

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+2, A, n)); polcoeff(A, n)}

Crossrefs

Cf. A195192, A205319.

Sequence in context: A348837 A349880 A260339 * A254243 A026025 A231375

Adjacent sequences: A205317 A205318 A205319 * A205321 A205322 A205323

Keyword

nonn

Author

Paul D. Hanna, Jan 25 2012

Status

approved