A176287 Diagonal sums of number triangle A092392.

1, 2, 7, 23, 81, 291, 1066, 3955, 14818, 55937, 212428, 810664, 3106167, 11942261, 46047897, 178000950, 689580319, 2676598447, 10406929687, 40525045518, 158022343991, 616950024334, 2411395005316, 9434753907065, 36948692202031

Offset

0,2

Comments

Hankel transform is A176288.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

G.f.: 1/(sqrt(1-4*x)*(1-x^2*c(x))) = 2/(sqrt(1-4*x)*(2-x+x*sqrt(1-4*x))), c(x) the g.f. of A000108.

a(n) = Sum_{k=0..floor(n/2)} C(2n-3k,n-k).

a(n) = Sum_{k=0..n} A000984(k)*A132364(n-k).

D-finite with recurrence: 2*n*a(n) +(6-11*n)*a(n-1) +(13*n-16)*a(n-2) +2*(5-n)*a(n-3) +3*(2-3*n)*a(n-4) +2*(2*n-5)*a(n-5)=0. - R. J. Mathar, Nov 15 2012 [Verified with Maple's FindRE and MinimalRecurrence functions, Georg Fischer, Nov 03 2022]

a(n) ~ 2^(2*n+3) / (7*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 04 2014

Maple

seq( add(binomial(2*n-3*k, n-k), k=0..floor(n/2)) , n=0..25); # G. C. Greubel, Nov 25 2019

Mathematica

CoefficientList[Series[2/(Sqrt[1-4*x]*(2-x+x*Sqrt[1-4*x])), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 04 2014 *)
a[n_]:= Sum[Binomial[2*n-3*k, n-k], {k, 0, Floor[n/2]}]; Table[a[n], {n, 0, 25}] (* G. C. Greubel, Oct 19 2016 *)

Prog

(PARI) a(n) = sum(k=0, n\2, binomial(2*n-3*k, n-k)); \\ Michel Marcus, Oct 20 2016
(Magma) [ &+[Binomial(2*n-3*k, n-k): k in [0..Floor(n/2)]] : n in [0..25]]; // G. C. Greubel, Nov 25 2019
(Sage) [sum(binomial(2*n-3*k, n-k) for k in (0..floor(n/2))) for n in (0..25)] # G. C. Greubel, Nov 25 2019
(GAP) List([0..25], n-> Sum([0..Int(n/2)], k-> Binomial(2*n-3*k, n-k) )); # G. C. Greubel, Nov 25 2019

Crossrefs

Cf. A000108, A000984, A092392, A132364.

Sequence in context: A130567 A091514 A143629 * A119371 A151290 A346627

Adjacent sequences: A176284 A176285 A176286 * A176288 A176289 A176290

Keyword

easy,nonn

Author

Paul Barry, Apr 14 2010

Status

approved