A153849 a(n) is the coefficient of x^n in the (n+2)-th self-composition of g.f. A(x) for n>=1, with a(1)=1.

1, 1, 4, 40, 675, 16023, 488264, 18150408, 798091737, 40691618165, 2372928554920, 156676126511294, 11619829396974162, 961567152709817189, 88268869225674860344, 8941327301073770038304, 994670849523023025795108

Offset

1,3

Links

Table of n, a(n) for n=1..17.

Example

G.f.: A(x) = x + x^2 + 4*x^3 + 40*x^4 + 675*x^5 + 16023*x^6 +...
Coefficients of successive iterations of A(x) are as follows:
[1,1,4,40,675,16023,488264,18150408,798091737,40691618165,...];
[1,2,10,101,1658,38327,1144412,41900873,1821202358,...];
[(1),3,18,189,3063,69204,2024472,72937181,3130859853,...];
[1,(4),28,310,5028,111648,3202916,113479712,4806652072,...];
[1,5,(40),470,7715,169475,4777592,166446886,6951781033,...];
[1,6,54,(675),11310,247443,6875376,235657813,9700212066,...];
[1,7,70,931,(16023),351372,9658544,326079187,13225766705,...];
[1,8,88,1244,22088,(488264),13331864,444121464,17753548560,...];
[1,9,108,1620,29763,666423,(18150408),597989364,23574131721,...];
[1,10,130,2065,39330,895575,24428084,(798091737),31060981566,...];
[1,11,154,2585,51095,1186988,32546888,1057515833,(40691618165),...];
where the diagonal of terms in parenthesis form this sequence.
Some explicit expansions:
A(A(x)) = x + 2*x^2 + 10*x^3 + 101*x^4 + 1658*x^5 + 38327*x^6 +...;
A(A(A(x))) = x + 3*x^2 + 18*x^3 + 189*x^4 + 3063*x^5 + 69204*x^6 +...;
A(A(A(A(x)))) = x + 4*x^2 + 28*x^3 + 310*x^4 + 5028*x^5 + 111648*x^6 +...

Prog

(PARI) {a(n)=local(F=x+x^2+sum(m=3, n-1, a(m)*x^m), G=x+x*O(x^n)); if(n<1, 0, if(n<=2, 1, for(i=1, n+1, G=subst(F, x, G)); return(polcoeff(G, n-1, x))))}

Crossrefs

Cf. A119819, A153389.

Sequence in context: A234294 A181088 A005431 * A372232 A251574 A010792

Adjacent sequences: A153846 A153847 A153848 * A153850 A153851 A153852

Keyword

nonn

Author

Paul D. Hanna, Jan 04 2009

Status

approved