A372522 G.f. A(x) satisfies A(A(A(A(A(A(x)))))) = Sum_{k>=1} k * 18^(k-1) * x^k.

0, 1, 6, -18, 378, -5670, 52488, 930204, -55108026, 575622774, 46483766460, -1494416264796, -85327731650772, 5947844644410876, 192190798316367540, -29067440301002581416, -418574641900663175706, 179341053539746099078422

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..200

Formula

Define the sequence b(n,m) as follows. If n<m, b(n,m) = 0, else if n=m, b(n,m) = 1, otherwise b(n,m) = 1/6 * ( 18^(n-m) * binomial(n+m-1,2*m-1) - Sum_{l=m+1..n-1} (b(n,l) + Sum_{k=l..n} (b(n,k) + Sum_{j=k..n} (b(n,j) + Sum_{i=j..n} (b(n,i) + Sum_{h=i..n} b(n,h) * b(h,i)) * b(i,j)) * b(j,k)) * b(k,l)) * b(l,m) ). a(n) = b(n,1).

Let F(x) = x/(1 - 18*x)^2, B(x) = A(A(x)) and C(x) = A(A(A(x))).

B(B(B(x))) = C(C(x)) = F(x).

B(x) = G(2*x)/2, where G(x) is the g.f. for A372499.

C(x) = H(9*x)/9, where H(x) is the g.f. for A309509.

Example

A(A(x)) = x + 12*x^2 + 36*x^3 + 432*x^4 - 62208*x^6 + 2846016*x^7 - ...
A(A(A(x))) = x + 18*x^2 + 162*x^3 + 1458*x^4 + 13122*x^5 + 2125764*x^7 + ...

Crossrefs

Cf. A309509, A372492, A372499, A372520.

Sequence in context: A354019 A052682 A214592 * A130437 A119813 A370716

Adjacent sequences: A372519 A372520 A372521 * A372523 A372524 A372525

Keyword

sign

Author

Seiichi Manyama, May 04 2024

Status

approved