A304642 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^(n+1) - A(x) )^n.

1, 2, 2, 10, 112, 1670, 30682, 663606, 16443254, 458349374, 14184612446, 482476888374, 17892738705864, 718662489646314, 31085968593760190, 1441017859748316954, 71281146361450601326, 3748236082140499881942, 208808936226479892694126, 12286084218797404915838902, 761413942238514103243322732, 49577303456014047226843229946, 3383829651598944830489407813422

Offset: 0

Author

Paul D. Hanna, May 16 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 2*x^2 + 10*x^3 + 112*x^4 + 1670*x^5 + 30682*x^6 + 663606*x^7 + 16443254*x^8 + 458349374*x^9 + 14184612446*x^10 + 482476888374*x^11 + ...
such that
1 = 1  +  ((1+x)^2 - A(x))  +  ((1+x)^3 - A(x))^2  +  ((1+x)^4 - A(x))^3  +  ((1+x)^5 - A(x))^4  +  ((1+x)^6 - A(x))^5  +  ((1+x)^7 - A(x))^6  +  ((1+x)^8 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1+x)^2/(1 + (1+x)*A(x))^2  +  (1+x)^6/(1 + (1+x)^2*A(x))^3  +  (1+x)^12/(1 + (1+x)^3*A(x))^4  +  (1+x)^20/(1 + (1+x)^4*A(x))^5  +  (1+x)^30/(1 + (1+x)^5*A(x))^6  +  (1+x)^42/(1 + (1+x)^6*A(x))^7 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A303056, A304639.

Programs

PARI

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

Formula

G.f. A(x) satisfies:

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

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

a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.16108865386542881383... and c = 0.154769618099522133628... - Vaclav Kotesovec, Oct 14 2020