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A260771 Certain directed lattice paths.

%I A260771 #27 Aug 19 2023 00:13:26
%S A260771 1,2,7,30,142,716,3771,20502,114194,648276,3737270,21819980,128757020,
%T A260771 766680856,4600866643,27797553638,168949310378,1032267189636,
%U A260771 6336728149794,39062959379620,241720286906116,1500910751651752,9348824475860702,58398701313158780
%N A260771 Certain directed lattice paths.
%C A260771 See Dziemianczuk (2014) for precise definition.
%H A260771 Lars Blomberg, <a href="/A260771/b260771.txt">Table of n, a(n) for n = 0..100</a>
%H A260771 M. Dziemianczuk, <a href="http://arxiv.org/abs/1410.5747">On Directed Lattice Paths With Additional Vertical Steps</a>, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
%F A260771 G.f.: P0(x) = 1/(1-x-x*P1(x)), where P1(x) = 2*(1-x)/(3*x) - 2*(sqrt(1-5*x-2*x^2)/(3*x))*sin(Pi/6 + arccos((20*x^3-6*x^2+15*x-2)/(2*(1-5*x-2*x^2)^(3/2)))/3). - See Dziemianczuk (2014), Proposition 11.
%F A260771 a(n) = Sum_{m=0..n+1} ((-1)^(n-m+1)*binomial(n+1,m) * Sum_{i=m..n+m} (binomial(i-1,i-m) * Sum_{j=0..n-m+1} (binomial(j,n+m-j-i)*2^(-n+m+2*j+i)*3^(n-m-j+1)*binomial(n-m+1,j))))/(n+1). - _Vladimir Kruchinin_, Feb 28 2016
%F A260771 a(n) ~ c * (22 + 10*sqrt(5))^(n/2) / n^(3/2), where c = 1/sqrt((5/2 - sqrt(5) + sqrt(85*sqrt(5)-190))*Pi) = 0.7820861193303307654051... . - _Vaclav Kotesovec_, Feb 28 2016
%t A260771 Table[Sum[(-1)^(n-l+1)*Binomial[n+1, l] * Sum[Binomial[i-1, i-l] * Sum[Binomial[j, n+l-j-i] * 2^(-n+l+2*j+i) * 3^(n-l-j+1) * Binomial[n-l+1, j], {j, 0, n-l+1}], {i, l, n+l}], {l, 0, n+1}]/(n+1), {n, 0, 20}] (* _Vaclav Kotesovec_, Feb 28 2016, after _Vladimir Kruchinin_ *)
%o A260771 (Maxima) a(n):=sum((-1)^(n-l+1)*binomial(n+1,l)*sum(binomial(i-1,i-l)*sum(binomial(j,n+l-j-i)*2^(-n+l+2*j+i)*3^(n-l-j+1)*binomial(n-l+1,j),j,0,n-l+1),i,l,n+l),l,0,n+1)/(n+1); /* _Vladimir Kruchinin_, Feb 28 2016 */
%K A260771 nonn
%O A260771 0,2
%A A260771 _N. J. A. Sloane_, Jul 30 2015
%E A260771 More terms from _Lars Blomberg_, Aug 01 2015