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

0,2

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.

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

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 *)

Prog

(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
(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
(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

Crossrefs

Sequence in context: A052961 A150662 A278391 * A150663 A054321 A344502

Adjacent sequences: A126565 A126566 A126567 * A126569 A126570 A126571

Keyword

nonn

Author

Philippe Deléham, Mar 13 2007

Status

approved