A317349 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - (1-x)^n )^n = 1.
1, 1, 2, 7, 42, 372, 4269, 59047, 946557, 17175289, 347208299, 7730688884, 187911183701, 4951155672353, 140575561645293, 4279249948000903, 139050095246322895, 4804391579357016747, 175902340755219278039, 6803436418471129704925, 277202774381386656583959, 11868116969794805874111831
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 42*x^4 + 372*x^5 + 4269*x^6 + 59047*x^7 + 946557*x^8 + 17175289*x^9 + 347208299*x^10 + ... such that 1 = 1 + (1/A(x) - (1-x)) + (1/A(x) - (1-x)^2)^2 + (1/A(x) - (1-x)^3)^3 + (1/A(x) - (1-x)^4)^4 + (1/A(x) - (1-x)^5)^5 + (1/A(x) - (1-x)^6)^6 + (1/A(x) - (1-x)^7)^7 + (1/A(x) - (1-x)^8)^8 + ... Also, A(x) = 1 + (1/A(x) - (1-x)^2) + (1/A(x) - (1-x)^3)^2 + (1/A(x) - (1-x)^4)^3 + (1/A(x) - (1-x)^5)^4 + (1/A(x) - (1-x)^6)^5 + (1/A(x) - (1-x)^7)^6 + (1/A(x) - (1-x)^8)^7 + (1/A(x) - (1-x)^9)^8 + ...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..300
Programs
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PARI
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = Vec( sum(m=0,#A, ( 1/Ser(A) - (1-x)^(m+1) )^m ) )[#A]/2 ); A[n+1]} for(n=0,25, print1(a(n),", "))
Formula
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^n )^n.
(2) A(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(n+1) )^n.
(3) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(n+1) )^n * (1-x)^(n+1).
(4) A(x)^2 = 2*A(x) * [ Sum_{n>=0} (n+1) * ( 1/A(x) - (1-x)^(n+1) )^n ] - [ Sum_{n>=0} (n+1) * ( 1/A(x) - (1-x)^(n+2) )^n ].
(5) A(x) = [ Sum_{n>=1} n*(n+1)/2 * (1-x)^(n+1) * ( 1/Ser(A) - (1-x)^(n+1) )^(n-1) ] / [ Sum_{n>=1} n^2 * (1-x)^n * ( 1/Ser(A) - (1-x)^n )^(n-1) ].
a(n) ~ 2^(log(2)/2 - 5/2) * n^n / (sqrt(1-log(2)) * exp(n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Aug 06 2018