A184884 Diagonal sums of number triangle A184883.

1, 1, 2, 6, 11, 27, 60, 132, 301, 669, 1502, 3370, 7543, 16919, 37912, 84968, 190457, 426841, 956698, 2144238, 4805827, 10771315, 24141588, 54108332, 121272549, 271806901, 609198390, 1365390546, 3060236911, 6858880431, 15372743856, 34454786384, 77223188593, 173079605553, 387921692082, 869445237846

Offset: 0

Author

Paul Barry, Jan 24 2011

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,2,-1,1).

Crossrefs

Cf. A183883.

Programs

Magma

A184883:= func< n,k | (&+[Binomial(k,j)*Binomial(2*(n-k), j)*2^j: j in [0..k]]) >;
A184884:= func< n | (&+[A184883(n, j): j in [0..Floor(n/2)]]) >;
[A184884(n): n in [0..40]]; // G. C. Greubel, Nov 19 2021

Mathematica

LinearRecurrence[{1,2,2,-1,1}, {1,1,2,6,11}, 45] (* G. C. Greubel, Nov 19 2021 *)

Sage

def A184883(n,k): return simplify( hypergeometric([-k, 2*(k-n)], [1], 2) )
def A184884(n): return sum( A184883(n, j) for j in (0..n//2) )
[A184884(n) for n in (0..40)] # G. C. Greubel, Nov 19 2021

Formula

G.f.: (1-x^2)/(1-x-2*x^2-2*x^3+x^4-x^5).

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..k} C(2*n-4*k,j)*C(k,j)*2^j.

a(n) = Sum_{k=0..floor(n/2)} Hypergeometric2F1([-k, 2*(k-n)], [1], 2). - G. C. Greubel, Nov 19 2021