|
%I #41 Sep 08 2022 08:45:57
%S 1,1,3,9,25,75,227,693,2139,6645,20757,65139,205189,648427,2054775,
%T 6526841,20775357,66251247,211617131,676930325,2168252571,6953348149,
%U 22322825865,71735559255,230735316795,742773456825,2392949225565,7714727440755,24888317247705,80341227688095
%N Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (1,2), and (2,1).
%H G. C. Greubel, <a href="/A191354/b191354.txt">Table of n, a(n) for n = 0..1000</a>
%H Steffen Eger, <a href="http://arxiv.org/abs/1511.00622">On the Number of Many-to-Many Alignments of N Sequences</a>, arXiv:1511.00622 [math.CO], 2015.
%F G.f.: 1/sqrt(1-2*x-3*x^2-4*x^3). - _Mark van Hoeij_, Apr 16 2013
%F G.f.: Q(0), where Q(k) = 1 + x*(2+3*x+4*x^2)*(4*k+1)/( 4*k+2 - x*(2+3*x+4*x^2)*(4*k+2)*(4*k+3)/(x*(2+3*x+4*x^2)*(4*k+3) + 4*(k+1)/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Sep 14 2013
%F a(n) = Sum_{k=0..n} (binomial(2*k,k) * Sum_{j=0..k} (binomial(j,n-k-j) *binomial(k,j)*2^(j-k)*3^(-n+k+2*j)*4^(n-k-2*j))). - _Vladimir Kruchinin_, Feb 27 2016
%F D-finite with recurrence: +(n)*a(n) +(-2*n+1)*a(n-1) +3*(-n+1)*a(n-2) +2*(-2*n+3)*a(n-3)=0. - _R. J. Mathar_, Jan 14 2020
%t a[n_]:= Sum[Binomial[2k, k]*Sum[Binomial[j, n-k-j]*Binomial[k, j]*2^(j-k) *3^(-n+k+2j)*4^(n-k-2j), {j, 0, k}], {k, 0, n}];
%t Array[a, 30, 0] (* _Jean-François Alcover_, Jul 21 2018, after _Vladimir Kruchinin_ *)
%t CoefficientList[Series[1/Sqrt[1-2*x-3*x^2-4*x^3], {x, 0, 30}], x] (* _G. C. Greubel_, Feb 18 2019 *)
%o (PARI) /* same as in A092566 but use */
%o steps=[[1,0], [1,1], [1,2], [2,1]];
%o /* _Joerg Arndt_, Jun 30 2011 */
%o (PARI) my(x='x+O('x^30)); Vec(1/sqrt(1-2*x-3*x^2-4*x^3)) \\ _G. C. Greubel_, Feb 18 2019
%o (Maxima)
%o a(n):=sum(binomial(2*k,k) * sum(binomial(j,n-k-j) * 2^(j-k) * binomial(k,j) * 3^(-n+k+2*j) * 4^(n-k-2*j),j,0,k),k,0,n); /* _Vladimir Kruchinin_, Feb 27 2016 */
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 1/Sqrt(1-2*x-3*x^2-4*x^3) )); // _G. C. Greubel_, Feb 18 2019
%o (Sage) (1/sqrt(1-2*x-3*x^2-4*x^3)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 18 2019
%Y Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369.
%K nonn
%O 0,3
%A _Joerg Arndt_, Jun 30 2011
|
