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A131765 Series reversion of x*(1-5x)/(1-x) .

%I A131765 #38 Sep 22 2024 04:27:58
%S A131765 1,4,36,404,5076,68324,963396,14046964,210062196,3204118724,
%T A131765 49656709476,779690085204,12376867734036,198301332087204,
%U A131765 3202580085625476,52080967814444724,852103170531254196,14016301507253656964
%N A131765 Series reversion of x*(1-5x)/(1-x) .
%C A131765 The Hankel transform of this sequence is 20^C(n+1,2).
%C A131765 a(n) is the number of small Schröder n-paths with 4 types of up steps (i.e., lattice paths from (0,0) to (2n,0) using steps U1=U2=U3=U4=(1,1), F=(2,0), D=(1,-1), with no F steps on the x-axis). - _Yu Hin Au_, Dec 05 2019
%H A131765 Vincenzo Librandi, <a href="/A131765/b131765.txt">Table of n, a(n) for n = 0..200</a>
%H A131765 Yu Hin (Gary) Au, <a href="https://arxiv.org/abs/1912.00555">Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers</a>, arXiv preprint, arXiv:1912.00555 [math.CO], 2019.
%F A131765 a(n) = Sum_{k=0..n} A086810(n,k)*4^k.
%F A131765 From _Paul Barry_, Sep 08 2009: (Start)
%F A131765 a(n) = Sum_{k=0..n} C(n+k,2*k)*A000108(k)*(-1)^(n-k)*5^k;
%F A131765 a(n) = Sum_{k=0..n} C(n+k,2*k)*A000108(k)*(4^(k+1)+(-1)^k)/5. (End)
%F A131765 Recurrence: (n+1)*a(n) = 9*(2*n-1)*a(n-1) - (n-2)*a(n-2) . - _Vaclav Kotesovec_, Oct 20 2012
%F A131765 a(n) ~ sqrt(40+18*sqrt(5))*(9+4*sqrt(5))^n/(10*sqrt(Pi)*n^(3/2)) . - _Vaclav Kotesovec_, Oct 20 2012. Equivalently, a(n) ~ phi^(6*n + 3) / (sqrt(2) * 5^(3/4) * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 07 2021
%F A131765 a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*hypergeom([k - n, n + 1], [k + 2], -4). - _Peter Luschny_, Jan 08 2018
%t A131765 Table[Sum[Binomial[n+k,2*k]*Binomial[2*k,k]/(k+1)*(-1)^(n-k)*5^k,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%t A131765 a[n_] := Sum[(-1)^k Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, -4], {k, 0, n}]; Table[a[n], {n, 0, 17}] (* _Peter Luschny_, Jan 08 2018 *)
%o A131765 (PARI) Vec(serreverse(x*(1-5*x)/(1-x) + O(x^30))) \\ _Michel Marcus_, Jan 08 2018
%Y A131765 Cf. A000108, A086810.
%K A131765 nonn
%O A131765 0,2
%A A131765 _Philippe Deléham_, Oct 29 2007
%E A131765 Extra terms added by _Paul Barry_, Sep 08 2009