A104970 Sum of squares of terms in even-indexed rows of triangle A104967.

1, 6, 18, 92, 298, 1444, 4852, 22840, 78490, 362580, 1265564, 5767688, 20366596, 91866984, 327351336, 1464522864, 5257011066, 23361650484, 84371466636, 372831130344, 1353477992556, 5952169844664, 21704580414936, 95051752387344

Offset

0,2

Comments

Sum of squares of terms in odd-indexed rows of triangle A104967 equals twice this sequence.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

G.f. A(x) satisfies: 2*(1+12*x)*A(x) - (1-16*x^2)*deriv(A(x), x) + 4 = 0.

a(n) = 2^(2*n-1)*(2 + Sum_{k=0..n-1} (-1)^k*binomial(2*k+1,k+1)/2^(2*k)). - Vaclav Kotesovec, Oct 28 2012

Mathematica

Flatten[{1, Table[2^(2*n-1)*(2+Sum[(-1)^k*Binomial[2*k+1, k+1]/2^(2*k), {k, 0, n-1}]), {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 28 2012 *)

Prog

(PARI) {a(n)=local(X=x+x*O(x^(2*n))); sum(k=0, 2*n, polcoeff(polcoeff((1-2*X)/(1-X-X*y*(1-2*X)), 2*n, x), k, y)^2)}
(Magma)
A104970:= func< n | n eq 0 select 1 else  4^n + (&+[(-1)^j*2^(2*n-2*j-1)*Binomial(2*j+1, j+1): j in [0..n-1]]) >;
[A104970(n): n in [0..40]]; // G. C. Greubel, Jun 09 2021
(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 A104970(n): return sum((A104967(2*n, k))^2 for k in (0..2*n))
[A104970(n) for n in (0..50)] # G. C. Greubel, Jun 09 2021

Crossrefs

Cf. A104967, A104968, A104969.

Sequence in context: A371987 A294471 A194995 * A216727 A174075 A151470

Adjacent sequences: A104967 A104968 A104969 * A104971 A104972 A104973

Keyword

nonn

Author

Paul D. Hanna, Mar 30 2005

Status

approved