A211180 G.f. satisfies: A(x) = x + A( x^2 + x^2*A(x) ).

1, 1, 1, 2, 3, 6, 10, 21, 40, 85, 170, 362, 752, 1618, 3438, 7447, 16065, 35080, 76574, 168424, 370880, 820968, 1820598, 4052594, 9038457, 20216002, 45301380, 101746560, 228918438, 516016266, 1165005168, 2634463663, 5965815375, 13528669545, 30717837778

Offset

1,4

Links

Paul D. Hanna, Table of n, a(n) for n = 1..512

Formula

Given g.f. A(x), define g(x) = x^2*(1 + A(x)); since A(x) = x + A(g(x))

then A(x) equals the sum of all iterations of g(x):

A(x) = x + g(x) + g(g(x)) + g(g(g(x))) + g(g(g(g(x)))) +...

Example

G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 10*x^7 + 21*x^8 +...
Define g(x) = x^2*(1 + A(x)) which satisfies: A(g(x)) = A(x) - x,
then the initial iterations of g(x) begin:
g(x) = x^2 + x^3 + x^4 + x^5 + 2*x^6 + 3*x^7 + 6*x^8 + 10*x^9 + 21*x^10 +...
g(g(x)) = x^4 + 2*x^5 + 4*x^6 + 7*x^7 + 14*x^8 + 26*x^9 + 52*x^10 +...
g(g(g(x))) = x^8 + 4*x^9 + 12*x^10 + 30*x^11 + 73*x^12 + 170*x^13 +...
g(g(g(g(x)))) = x^16 + 8*x^17 + 40*x^18 + 156*x^19 + 530*x^20 +...
where A(x) = x + g(x) + g(g(x)) + g(g(g(x))) + g(g(g(g(x)))) +...

Prog

(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x+subst(A, x, x^2*(1+A)+x*O(x^n))); polcoeff(A, n)}
for(n=1, 45, print1(a(n), ", "))

Crossrefs

Sequence in context: A359019 A002988 A138347 * A265582 A242563 A240513

Adjacent sequences: A211177 A211178 A211179 * A211181 A211182 A211183

Keyword

nonn

Author

Paul D. Hanna, May 11 2012

Status

approved