A320951 G.f.: A(x) satisfies: A(x) = Sum_{n>=0} x^n * (1+x)^(n*(n+1)) / A(x)^n.

1, 1, 2, 3, 9, 28, 110, 485, 2358, 12486, 70726, 425747, 2702837, 18004835, 125337381, 908737863, 6843536374, 53407750147, 431075414218, 3592384229312, 30862831600689, 272976843937138, 2482698463801148, 23192576636266041, 222310388884578760, 2184486850658804107, 21985733344615744620, 226455749821063728474, 2385331864619907236147, 25676170688883138634306, 282253492062060457638824

Offset: 0

Author

Paul D. Hanna, Nov 20 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 9*x^4 + 28*x^5 + 110*x^6 + 485*x^7 + 2358*x^8 + 12486*x^9 + 70726*x^10 + 425747*x^11 + 2702837*x^12 + ...
such that
A(x) = 1 + x*(1+x)^2/A(x) + x^2*(1+x)^6/A(x)^2 + x^3*(1+x)^12/A(x)^3 + x^4*(1+x)^20/A(x)^4 + x^5*(1+x)^30/A(x)^5 + ...
Also
1 + x = 1 + x/A(x) + x^2*(1+x)^2/A(x)^2 + x^3*(1+x)^6/A(x)^3 + x^4*(1+x)^12/A(x)^4 + x^5*(1+x)^20/A(x)^5 + x^6*(1+x)^30/A(x)^6 + ...
RELATED SERIES.
Sum_{n>=0} x^n * (1+x)^(n^2) / A(x)^n = 1 + x + x^2 + x^3 + 3*x^4 + 8*x^5 + 32*x^6 + 135*x^7 + 649*x^8 + 3381*x^9 + 18894*x^10 + 112382*x^11 + 705174*x^12 + ...
A(A(x)-1) = 1 + x + 4*x^2 + 14*x^3 + 56*x^4 + 251*x^5 + 1239*x^6 + 6627*x^7 + 38112*x^8 + 233692*x^9 + 1517788*x^10 + 10384824*x^11 + ...
where A(A(x)-1) = Sum_{n>=0} (A(x)-1)^n * A(x)^(n*(n+1)) / A(A(x)-1)^n.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A318644, A303058.

Programs

PARI

{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec(sum(n=0, #A, ((1+x)^n +x*O(x^#A))^(n+1) * x^n/Ser(A)^n ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:

(1) A(x) = Sum_{n>=0} x^n * (1+x)^(n*(n+1)) / A(x)^n.

(2) 1 + x = Sum_{n>=0} x^n * (1+x)^(n*(n-1)) / A(x)^n.

(3) A(x) = Sum_{n>=0} (A(x)-1)^n * A(x)^(n*(n-1)) / A(A(x)-1)^n.