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

Author

Paul Barry, Apr 14 2010

Keywords

• easy
• nonn

Comments

Hankel transform is A176288.

Links

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

Crossrefs

Cf. A000108, A000984, A092392, A132364.

Programs

GAP

List([0..25], n-> Sum([0..Int(n/2)], k-> Binomial(2*n-3*k, n-k) )); # G. C. Greubel, Nov 25 2019

Magma

[ &+[Binomial(2*n-3*k, n-k): k in [0..Floor(n/2)]] : n in [0..25]]; // G. C. Greubel, Nov 25 2019

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 *)

PARI

a(n) = sum(k=0, n\2, binomial(2*n-3*k, n-k)); \\ Michel Marcus, Oct 20 2016

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

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