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Binomial transform of A026641.
7

%I #24 Nov 02 2023 10:01:58

%S 1,2,7,29,127,572,2623,12182,57115,269750,1281457,6116585,29310721,

%T 140925176,679493983,3284357789,15909178627,77208716606,375330428293,

%U 1827310839359,8908332730957,43481990059796,212472526927393

%N Binomial transform of A026641.

%C The Hankel transform of this sequence is 3^n (see A000244).

%C Row sums of triangle in A110877. - _Philippe Deléham_, Oct 10 2007

%H Vincenzo Librandi, <a href="/A126568/b126568.txt">Table of n, a(n) for n = 0..200</a>

%H Isaac DeJager, Madeleine Naquin, Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.

%F a(n) = Sum_{0<=k<=n} A110877(n,k). - _Philippe Deléham_, Oct 10 2007

%F Conjecture: 4*n*a(n) +2*(2*n-7)*a(n-1) +(-163*n+267)*a(n-2) +10*(23*n-58)*a(n-3) +75*(-n+3)*a(n-4) = 0. - _R. J. Mathar_, Jun 30 2013

%F G.f.: (11*x^4 -12*x^3 -x^2 +3*x -1 -sqrt(5*x^2-6*x+1)*(5*x^3-3*x^2-1))/( sqrt(5*x^2-6*x+1)*(4*x^4-8*x^3-3*x^2+7*x-2) -10*x^5 +32*x^4 -31*x^3 + 20*x^2 -13*x +2). - _Vladimir Kruchinin_, Apr 08 2014

%F a(n) ~ 5^(n + 1/2) / (3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Nov 02 2023

%t CoefficientList[Series[-(-11x^4 +Sqrt[5x^2-6x+1](5x^3-3x^2-1) +12x^3+x^2 -3x+1)/(-10x^5 +Sqrt[5x^2-6x+1](4x^4-8x^3-3x^2+7x-2) +32x^4-31 x^3+20x^2 -13x+2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Apr 09 2014 *)

%o (PARI) my(x='x+O('x^30)); Vec((11*x^4 -12*x^3 -x^2 +3*x -1 -sqrt(5*x^2 -6*x +1)*(5*x^3-3*x^2-1))/( sqrt(5*x^2-6*x+1)*(4*x^4-8*x^3-3*x^2+7*x-2) -10*x^5 +32*x^4 -31*x^3 + 20*x^2 -13*x +2)) \\ _G. C. Greubel_, Feb 15 2019

%o (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (11*x^4-12*x^3-x^2+3*x-1 -Sqrt(5*x^2-6*x+1)*(5*x^3-3*x^2-1))/( Sqrt(5*x^2-6*x+1)*(4*x^4-8*x^3-3*x^2+7*x-2) -10*x^5+32*x^4-31*x^3+ 20*x^2-13*x+2) )); // _G. C. Greubel_, Feb 15 2019

%o (Sage) m=30; a=((11*x^4-12*x^3-x^2+3*x-1 -sqrt(5*x^2-6*x+1)*(5*x^3-3*x^2-1))/( sqrt(5*x^2-6*x+1)*(4*x^4-8*x^3-3*x^2+7*x-2) -10*x^5+32*x^4-31*x^3 + 20*x^2-13*x+2)).series(x, m+2).coefficients(x, sparse=False); a[0:m] # _G. C. Greubel_, Feb 15 2019

%K nonn

%O 0,2

%A _Philippe Deléham_, Mar 13 2007