A104969 Sum of squares of terms in rows of triangle A104967.

1, 2, 6, 12, 18, 36, 92, 184, 298, 596, 1444, 2888, 4852, 9704, 22840, 45680, 78490, 156980, 362580, 725160, 1265564, 2531128, 5767688, 11535376, 20366596, 40733192, 91866984, 183733968, 327351336, 654702672, 1464522864, 2929045728

Offset: 0

Author

Paul D. Hanna, Mar 30 2005

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A104967, A104968, A104970.

Programs

Mathematica

A104967[n_, k_]:= A104967[n, k]= Sum[(-2)^j*Binomial[k+1, j]*Binomial[n-j, k], {j, 0, n-k}];
A104969[n_]:= A104969[n]= Sum[A104967[n, k]^2, {k,0,n}];
Table[A104969[n], {n, 0, 50}] (* G. C. Greubel, Jun 09 2021 *)

PARI

{a(n)=local(X=x+x*O(x^n)); sum(k=0,n,polcoeff(polcoeff((1-2*X)/(1-X-X*y*(1-2*X)),n,x),k,y)^2)}

Sage

@cached_function
def A104967(n,k): return sum( (-2)^j*binomial(k+1,j)*binomial(n-j,k) for j in (0..n-k))
def A104969(n): return sum((A104967(n,k))^2 for k in (0..n))
[A104969(n) for n in (0..50)] # G. C. Greubel, Jun 09 2021

Formula

a(2*n+1) = 2*a(2*n).

G.f.: A(x) = (1+2*x)*G(x^2) where G(x) is the g.f. of A104970 such that G(x) satisfies: 2*(1+12*x)*G(x) - (1-16*x^2)*deriv(G(x), x) + 4 = 0.

a(n) = Sum_{k=0..n} (A104967(n, k))^2.