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%I #46 Mar 18 2023 15:20:27
%S 1,10,76,568,4240,31648,236224,1763200,13160704,98232832,733219840,
%T 5472827392,40849739776,304906608640,2275853910016,16987204845568,
%U 126794223124480,946404965613568,7064062832410624,52726882796830720
%N Generalized NSW numbers.
%C Counts total area under elevated Schroeder paths of length 2n+2, where horizontal steps can choose from three colors.
%C Case r=3 for family (1+(r-1)x)/(1-2(1+r)x+(1-r)^2*x^2). Case r=2 gives NSW numbers A002315 and case r=4 gives NSW numbers A096053.
%C Fifth binomial transform of (1+8x)/(1-16x^2), A107906.
%C If p is an odd prime, a((p-1)/2) == 1 mod p. - _Altug Alkan_, Mar 17 2016
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-4).
%F G.f.: (1+2*x)/(1-8*x+4*x^2). [corrected by _Ralf Stephan_, Nov 30 2010]
%F a(n) = Sum_{k=0..n} binomial(2*n+1, 2*k)*3^k.
%F a(n) = ((1+sqrt(3))*(4+2*sqrt(3))^n+(1-sqrt(3))*(4-2*sqrt(3))^n)/2 = A099156(n+1)+2*A099156(n).
%F a(n) = 8*a(n-1) - 4*a(n-2); a(0) = 1, a(1) = 10. - _Lekraj Beedassy_, Apr 19 2020
%F a(n) = 2^n*A001834(n). - _Philippe Deléham_, Mar 18 2023
%t Table[Sum[Binomial[2 n + 1, 2 k] 3^k, {k, 0, n}], {n, 0, 20}] (* or *) CoefficientList[Series[(1 + 2 x)/(1 - 8 x + 4 x^2), {x, 0, 20}], x] (* _Michael De Vlieger_, Mar 17 2016 *)
%o (PARI) Vec((1+2*x)/(1-8*x+4*x^2) + O(x^40)) \\ _Michel Marcus_, Mar 17 2016
%Y Cf. A001834, A099156, A102591.
%K easy,nonn
%O 0,2
%A _Paul Barry_, May 27 2005
%E Typo corrected and link added by _Johannes W. Meijer_, Aug 07 2010
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