A195193 G.f.: A(x) = x + x*ITERATE^2(x + 2*x*ITERATE^2(x + 3*x*ITERATE^2(x + 4*x*ITERATE^2(x + ...)))), where ITERATE^2(F(x)) = F(F(x)), and the nested iterations continue indefinitely.

1, 1, 4, 32, 392, 6492, 135876, 3450048, 103437240, 3592136052, 142420085352, 6373515782544, 318952925294496, 17709716546754768, 1083721599571837632, 72661988008040494752, 5310638869813310865984, 421165309915181979699168, 36095229917026934599948992

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..100

Example

G.f.: A(x) = x + x^2 + 4*x^3 + 32*x^4 + 392*x^5 + 6492*x^6 +...
where A(x) is generated by nested iterations of shifted series:
A(x) = x + x*B(B(x)), where
B(x) = x + 2*x^2 + 12*x^3 + 132*x^4 + 2118*x^5 + 44400*x^6 +...;
B(x) = x + 2*x*C(C(x)), where
C(x) = x + 3*x^2 + 24*x^3 + 336*x^4 + 6672*x^5 + 169560*x^6 +...;
C(x) = x + 3*x*D(D(x)), where
D(x) = x + 4*x^2 + 40*x^3 + 680*x^4 + 16100*x^5 + 481200*x^6 +...;
D(x) = x + 4*x*E(E(x)), where
E(x) = x + 5*x^2 + 60*x^3 + 1200*x^4 + 33000*x^5 + 1134420*x^6 +...;
E(x) = x + 5*x*F(F(x)), where
F(x) = x + 6*x^2 + 84*x^3 + 1932*x^4 + 60522*x^5 + 2352672*x^6 +...;
E(x) = x + 6*x*G(G(x)), where
G(x) = x + 7*x^2 + 112*x^3 + 2912*x^4 + 102368*x^5 + 4440240*x^6 +...;
G(x) = x + 7*x*H(H(x)), where
H(x) = x + 8*x^2 + 144*x^3 + 4176*x^4 + 162792*x^5 + 7794720*x^6 +...; ...
Also, the 2nd iteration of the g.f. A(x) begins:
A(A(x)) = x + 2*x^2 + 10*x^3 + 85*x^4 + 1044*x^5 + 16996*x^6 + 347960*x^7 + 8649376*x^8 + 254561796*x^9 + 8704440152*x^10 +...

Prog

(PARI) {a(n)=local(A=x); for(k=0, n, A=x + (n-k+1)*x*subst(A, x, A+x*O(x^n))); polcoeff(A, n)}

Crossrefs

Sequence in context: A222412 A349601 A007763 * A203435 A349558 A005263

Adjacent sequences: A195190 A195191 A195192 * A195194 A195195 A195196

Keyword

nonn

Author

Paul D. Hanna, Sep 11 2011

Status

approved