A182964 G.f.: x = Sum_{n>=1} a(n)*G_n(x)^n where G_n(x) is the n-th iteration of (x-x^2).

1, 1, 4, 28, 278, 3558, 55600, 1024459, 21721922, 520585084, 13908591504, 409749226342, 13192559204352, 460792929469356, 17351608517036848, 700669956538570048, 30200986529461293238, 1383912736505236230958

Offset

1,3

Comments

Compare g.f. of this sequence to the g.f. of A000272(n) = n^(n-2):

. x = Sum_{n>=1} n^(n-2)*[x/(1+n*x)]^n

where x/(1+n*x) = n-th iteration of x/(1+x).

Links

Table of n, a(n) for n=1..18.

Example

G.f.: x = 1*G_1(x) + 1*G_2(x)^2 + 4*G_3(x)^3 + 28*G_4(x)^4 + 278*G_5(x)^5 + 3558*G_6(x)^6 + 55600*G_7(x)^7 +...+ a(n)*G_n(x)^n +...
where the initial iterations of G(x) = (x-x^2) begin:
G_1(x) = x - x^2;
G_2(x) = x - 2*x^2 + 2*x^3 - x^4;
G_3(x) = x - 3*x^2 + 6*x^3 - 9*x^4 + 10*x^5 - 8*x^6 + 4*x^7 - x^8;
G_4(x) = x - 4*x^2 + 12*x^3 - 30*x^4 + 64*x^5 - 118*x^6  +-...;
G_5(x) = x - 5*x^2 + 20*x^3 - 70*x^4 + 220*x^5 - 630*x^6 +-...;
...
Coefficients in the n-th power of the n-th iteration of x-x^2 begin:
G_1(x)^1: [1, -1, 0, 0, 0, 0, 0, ...];
G_2(x)^2: [1, -4, 8, -10, 8, -4, 1, 0, 0, 0, 0, 0, ...];
G_3(x)^3: [1, -9, 45, -162, 462, -1095, 2217, -3900, 6024, ...];
G_4(x)^4: [1, -16, 144, -952, 5120, -23608, 96296, -354664, ...];
G_5(x)^5: [1, -25, 350, -3600, 30225, -218775, 1410780, ...];
G_6(x)^6: [1, -36, 720, -10530, 125400, -1286376, 11759727, ...];
...

Prog

(PARI) /* n-th Iteration of a function: */
{ITERATE(n, F, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}
/* G.f.: */
{a(n)=local(G=x); if(n<=1, G=x, G=x-sum(k=1, n-1, a(k)*ITERATE(k, x-x^2, n)^k)); polcoeff(G, n)}

Crossrefs

Sequence in context: A128318 A032274 A374601 * A306228 A178599 A369088

Adjacent sequences: A182961 A182962 A182963 * A182965 A182966 A182967

Keyword

nonn

Author

Paul D. Hanna, Jan 01 2011

Status

approved