A240434 Binomial transform of the sum of the first n even squares (A002492).

0, 4, 28, 128, 480, 1600, 4928, 14336, 39936, 107520, 281600, 720896, 1810432, 4472832, 10895360, 26214400, 62390272, 147062784, 343670784, 796917760, 1835008000, 4198498304, 9550430208, 21609054208, 48653926400, 109051904000, 243403849728, 541165879296

Offset: 0

Author

Wesley Ivan Hurt, Apr 04 2014

Keywords

• nonn

Comments

The inverse binomial transform of a(n) is A002492(n) = 2n(n+1)(2n+1)/3.

Crossrefs

Cf. A002492.

Programs

Mathematica

Table[Sum[2 Binomial[n, k] k (k + 1) (2 k + 1)/3, {k, 0, n}], {n, 0, 30}]

Formula

Conjecture: a(n) = (2^(n-1)*n*(5+6*n+n^2))/3. G.f.: -4*x*(x-1) / (2*x-1)^4. - Colin Barker, Apr 06 2014

a(n) = (-1)^n * Sum_{k=0..floor(n/2)} binomial(n-k, k) * (-4)^(n-k) * (n-k). - Joseph M. Shunia, Jul 20 2022