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A186185 Expansion of 1/(1 - x*A001764(x/(1-x))/(1-x)). 0

%I

%S 1,1,3,11,48,239,1306,7612,46436,292875,1894365,12496864,83753165,

%T 568628232,3902600850,27031069848,188709211952,1326456525471,

%U 9379857716098,66680723764051,476269444919163,3416178576731504

%N Expansion of 1/(1 - x*A001764(x/(1-x))/(1-x)).

%H Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.

%H Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.

%F a(n) = Sum_{m=1..n} Sum_{k=m..n} binomial(n-1,k-1)*m/(2*k-m)*binomial(3*k-2*m-1,k-m), n>0, a(0)=1.

%e G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 48*x^4 + 239*x^5 + ...

%e The g.f. of A001764, where A001764(x) = 1 + x*A001764(x)^3, begins:

%e A001764(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + ...

%e The g.f. of the binomial transform of A001764 begins:

%e A001764(x/(1-x))/(1-x) = 1 + 2*x + 6*x^2 + 25*x^3 + 126*x^4 + 704*x^5 + ...

%e where A(x) = 1/(1 - x*A001764(x/(1-x))/(1-x)).

%o (PARI) {a(n)=if(n<0,0,if(n==0,1,sum(m=1,n,sum(k=m,n,binomial(n-1,k-1)*binomial(3*k-2*m-1,k-m)*m/(2*k-m)))))}

%o (PARI) {a(n)=local(A001764=sum(m=0,n,binomial(3*m,m)*x^m/(2*m+1))+O(x^n));polcoeff(1/(1-x*subst(A001764,x,x/(1-x))/(1-x)),n)}

%K nonn

%O 0,3

%A _Vladimir Kruchinin_, Feb 14 2011

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Last modified September 26 08:44 EDT 2021. Contains 347664 sequences. (Running on oeis4.)