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A229548 E.g.f. satisfies: A(x) = x + A'(x) * Integral A(x) dx. 1

%I

%S 1,1,4,32,412,7664,193080,6308664,259111920,13063094736,793227096672,

%T 57121999975104,4815054009921024,469833060580971072,

%U 52555992910144441344,6682566041846191689984,958596887629217123483904,154093408614272498448416256,27591229509407916324655477248

%N E.g.f. satisfies: A(x) = x + A'(x) * Integral A(x) dx.

%H Vaclav Kotesovec, <a href="/A229548/b229548.txt">Table of n, a(n) for n = 1..200</a>

%F a(n) = Sum_{k=1..n-1} binomial(n,k-1)*a(n-k)*a(k) for n>1 with a(1)=1.

%F a(n) ~ c * (n!)^2 / (2^n * n^(1/3)), where c = 3.081214203431821156695905553610151693827575050546... - _Vaclav Kotesovec_, Feb 20 2014

%e E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 32*x^4/4! + 412*x^5/5! + 7664*x^6/6! +...

%e such that A(x) = x + A'(x)*B(x) where

%e A'(x) = 1 + x + 4*x^2/2! + 32*x^3/3! + 412*x^4/4! + 7664*x^5/5! +...

%e B(x) = x^2/2! + x^3/3! + 4*x^4/4! + 32*x^5/5! + 412*x^6/6! + 7664*x^7/7! +...

%e so that B(x) = Integral A(x) dx (here integration does not include constant term).

%t a = ConstantArray[0,20]; a[[1]]=1; Do[a[[n]] = Sum[Binomial[n,k-1]*a[[n-k]]*a[[k]],{k,1,n-1}],{n,2,20}]; a (* _Vaclav Kotesovec_, Feb 19 2014 *)

%o (PARI) {a(n)=local(A=x+x^2);for(i=1,n,A=x+A'*intformal(A+x*O(x^n)));n!*polcoeff(A,n)}

%o for(n=1,20,print1(a(n),", "))

%o (PARI) /* Recurrence: */

%o {a(n)=if(n==1,1,sum(k=1,n-1,binomial(n,k-1)*a(n-k)*a(k)))}

%o for(n=1,20,print1(a(n),", "))

%K nonn

%O 1,3

%A _Paul D. Hanna_, Sep 26 2013

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Last modified September 16 20:51 EDT 2021. Contains 347473 sequences. (Running on oeis4.)