A245389 G.f. satisfies: A(x) = Sum_{n>=0} x^n / (1 - (n+1)*x*A(x)).
1, 2, 6, 23, 102, 496, 2570, 13959, 78682, 457243, 2727360, 16647048, 103759186, 659500772, 4271197824, 28175622291, 189321228022, 1296246842443, 9049626101836, 64481397834665, 469461395956168, 3497006117588399, 26688813841105524, 208977790442594368, 1680981707733908594
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 2*x + 6*x^2 + 23*x^3 + 102*x^4 + 496*x^5 + 2570*x^6 +... where we have the following series identity: A(x) = 1/(1-x*A(x)) + x/(1-2*x*A(x)) + x^2/(1-3*x*A(x)) + x^3/(1-4*x*A(x)) + x^4/(1-5*x*A(x)) + x^5/(1-6*x*A(x)) + x^6/(1-7*x*A(x)) +... is equal to A(x) = 1/(1-x) + x/(1-x)^2*A(x)/(1+x*A(x)) + 2!*x^2/(1-x)^3*A(x)^2/((1+x*A(x))*(1+2*x*A(x))) + 3!*x^3/(1-x)^4*A(x)^3/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))) + 4!*x^4/(1-x)^5*A(x)^4/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))*(1+4*x*A(x))) + 5!*x^5/(1-x)^6*A(x)^5/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))*(1+4*x*A(x))*(1+5*x*A(x))) +...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..300
Programs
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PARI
{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0, n, x^m/(1-(m+1)*x*A+x*O(x^n))));polcoeff(A, n)} for(n=0,30,print1(a(n),", ")) -
PARI
{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0, n, m!*x^m*A^m/(1-x +x*O(x^n))^(m+1)/prod(k=1, m, 1+k*x*A +x*O(x^n))));polcoeff(, n)} for(n=0,30,print1(a(n),", "))
Formula
G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n / (1 - (n+1)*x*A(x)).
(2) A(x) = Sum_{n>=0} n! * x^n/(1-x)^(n+1) * A(x)^n / Product_{k=1..n} (1 + k*x*A(x)).