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A191649 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (1,1), (2,2).

%I A191649 #34 Apr 18 2025 13:12:38
%S A191649 1,3,14,71,379,2082,11651,66051,378064,2180037,12644861,73695358,
%T A191649 431209313,2531556197,14904832196,87970766447,520337606401,
%U A191649 3083584244460,18304476242735,108820740004749,647817646760368,3861215365595659,23039691494489015,137615812845579390
%N A191649 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (1,1), (2,2).
%H A191649 G. C. Greubel, <a href="/A191649/b191649.txt">Table of n, a(n) for n = 0..1000</a>
%H A191649 Paul Barry, <a href="https://arxiv.org/abs/2504.09719">Notes on Riordan arrays and lattice paths</a>, arXiv:2504.09719 [math.CO], 2025. See pp. 9, 29.
%F A191649 G.f.: 1/sqrt(x^4 +2*x^3 -x^2 -6*x +1). - _Mark van Hoeij_, Apr 17 2013
%F A191649 D-finite with recurrence: n*a(n) +3*(-2*n+1)*a(n-1) +(-n+1)*a(n-2) +(2*n-3)*a(n-3) +(n-2)*a(n-4)=0. - _R. J. Mathar_, Oct 08 2016
%t A191649 CoefficientList[Series[1/Sqrt[x^4 + 2 x^3 - x^2 - 6 x + 1], {x, 0, 23}], x] (* _Michael De Vlieger_, Oct 08 2016 *)
%o A191649 (PARI) /* same as in A092566 but use */
%o A191649 steps=[[0,1], [1,0], [1,1], [2,2]];
%o A191649 /* _Joerg Arndt_, Jun 30 2011 */
%o A191649 (PARI) my(x='x+O('x^30)); Vec(1/sqrt(x^4+2*x^3-x^2-6*x+1)) \\ _G. C. Greubel_, Apr 29 2019
%o A191649 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(x^4+2*x^3-x^2-6*x+1) )); // _G. C. Greubel_, Apr 29 2019
%o A191649 (Sage) (1/sqrt(x^4+2*x^3-x^2-6*x+1)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 29 2019
%Y A191649 Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369, A191354.
%K A191649 nonn
%O A191649 0,2
%A A191649 _Joerg Arndt_, Jun 30 2011