A153850 a(n) is the coefficient of x^(2*n-1) in the n-fold self-composition of G(x) = x + G(G(x))^3 = g.f. of A153851.

1, 2, 27, 594, 17180, 603879, 24795645, 1160887350, 60940292571, 3541938123306, 225669592036086, 15634133444509443, 1169781625911185118, 93989088711427170141, 8069678384570571946581, 737204558292074214218778

Offset

1,2

Links

Table of n, a(n) for n=1..16.

Example

Let A(x) be the g.f. of A153851, which begins
A(x) = x + x^3 + 6*x^5 + 57*x^7 + 683*x^9 + 9474*x^11 + 145815*x^13 + 2430393*x^15 + 43202448*x^17 + ... + A153851(2*n-1)*x^n + ...
then A(x) satisfies A(x - A(x)^3) = x.
Further, let successive iterations of A(x) be denoted by
B(x) = A(A(x)) = g.f. of A153852,
C(x) = A(A(A(x))) = g.f. of A153853,
D(x) = A(A(A(A(x)))) = g.f. of A153854, etc.,
then the nonzero coefficients in the successive iterations of A(x) form the table:
A:[1, 1, 6, 57, 683, 9474, 145815, 2430393, ...];
B:[1, 2, 15, 165, 2213, 33693, 561867, 10053141, ...];
C:[1, 3, 27, 339, 5067, 84738, 1536867, 29687772, ...];
D:[1, 4, 42, 594, 9827, 179928, 3545637, 73988631, ...];
E:[1, 5, 60, 945, 17180, 342765, 7316178, 164606166, ...];
F:[1, 6, 81, 1407, 27918, 603879, 13907133, 336334443, ...];
G:[1, 7, 105, 1995, 42938, 1001973, 24795645, 642380025, ...];
H:[1, 8, 132, 2724, 63242, 1584768, 41975610, 1160887350, ...]; ...
in which the main diagonal equals this sequence.

Prog

(PARI) {a(n) = my(G=x+O(x^(2*n+1)), H=G); for(i=0, n, G=serreverse(x-G^3)); for(i=1, n, H=subst(G, x, H)); polcoeff(H, 2*n-1)}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A153851, A153852, A153853, A153854.

Sequence in context: A242009 A121137 A203429 * A138458 A090248 A300591

Adjacent sequences: A153847 A153848 A153849 * A153851 A153852 A153853

Keyword

nonn

Author

Paul D. Hanna, Jan 21 2009

Status

approved