A126568 Binomial transform of A026641.

1, 2, 7, 29, 127, 572, 2623, 12182, 57115, 269750, 1281457, 6116585, 29310721, 140925176, 679493983, 3284357789, 15909178627, 77208716606, 375330428293, 1827310839359, 8908332730957, 43481990059796, 212472526927393

Offset: 0

Author

Philippe Deléham, Mar 13 2007

Keywords

• nonn

Comments

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

Row sums of triangle in A110877. - Philippe Deléham, Oct 10 2007

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Programs

Magma

m:=25; R:=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

Mathematica

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 *)

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

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

Formula

a(n) = Sum_{0<=k<=n} A110877(n,k). - Philippe Deléham, Oct 10 2007

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

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

a(n) ~ 5^(n + 1/2) / (3*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 02 2023