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E.g.f.: exp(x*C(x)^2) where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers, A000108.
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%I #45 Jun 12 2024 01:14:09

%S 1,1,5,43,529,8501,169021,4010455,110676833,3484717129,123320412181,

%T 4847038223171,209536628422705,9882471447634813,505033804901100749,

%U 27802319803528367791,1640388588050579832001,103275015543414629215505,6910877628962983581031333

%N E.g.f.: exp(x*C(x)^2) where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers, A000108.

%H G. C. Greubel, <a href="/A251568/b251568.txt">Table of n, a(n) for n = 0..360</a>

%F a(n) = Sum_{k=0..n} (n!/k!) * binomial(2*n-1, n-k) * 2*k/(n+k) for n > 0 with a(0)=1.

%F E.g.f. A(x) satisfies: A'(x)/A(x) = C'(x) = C(x)^2 / sqrt(1-4*x) where C(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function.

%F Recurrence equation: a(n) = -(n^2 - 5*n +1)*a(n-1) + n*(2*n - 3)*(2*n - 4)*a(n-2) with a(0) = 1, a(1) = 1. It appears that a(n) - 1 is divisible by n*(n - 1) for n >= 2. - _Peter Bala_, Feb 14 2015

%F a(n) ~ 2^(2*n+1/2) * n^(n-1) / exp(n-1). - _Vaclav Kotesovec_, Feb 14 2015

%F a(n) are special values of the hypergeometric function 1F1: a(n) = 4^n*Gamma(n+1/2)*exp(-1)*hypergeom([2*n+1], [n+2], 1)/(sqrt(Pi)*(n+1)), for n>=1. - _Karol A. Penson_, Jun 01 2015

%F a(n) = ((2*n)!/(n+1)!)*hypergeometric([1-n],[n+2],-1), a(0)=1. - _Vladimir Kruchinin_, May 03 2017

%e E.g.f.: A(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 529*x^4/4! + 8501*x^5/5! + ...

%e where

%e log(A(x)) = x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + ... + A000108(n)*x^n + ...

%p CatalanNumber := n -> binomial(2*n,n)/(n+1):

%p a := n -> `if`(n=0, 1, n!*CatalanNumber(n)*hypergeom([1-n], [2+n], -1)):

%p seq(simplify(a(n)), n=0..9); # _Peter Luschny_, May 04 2017

%t Flatten[{1,Table[Sum[n!/k!*Binomial[2*n-1,n-k]*2*k/(n+k),{k,1,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Feb 14 2015 *)

%t a[0] = 1; a[n_] := (2n)!/(n+1)! Hypergeometric1F1[1-n, n+2, -1];

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 03 2017, after _Vladimir Kruchinin_ *)

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

%o for(n=0,20,print1(a(n),", "))

%o (PARI) {a(n) = if(n==0, 1, sum(k=1, n, n!/k! * binomial(2*n-1, n-k) * 2*k/(n+k) ))}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A080893, A001517, A251569, A000108.

%K nonn,easy

%O 0,3

%A _Paul D. Hanna_, Dec 05 2014