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A337467 Expansion of sqrt(2 / ( (1-2*x+49*x^2) * (1-7*x+sqrt(1-2*x+49*x^2)) )). 2

%I #20 Apr 29 2021 04:32:33

%S 1,3,-21,-139,531,6489,-9723,-292293,-135117,12514313,29905809,

%T -501239553,-2310673379,18245192679,140574917259,-562805403867,

%U -7557237645741,11275709877369,371974318253601,201852054629631,-16932135947326551,-42530838930147813,709138646702505999

%N Expansion of sqrt(2 / ( (1-2*x+49*x^2) * (1-7*x+sqrt(1-2*x+49*x^2)) )).

%H Seiichi Manyama, <a href="/A337467/b337467.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(2*k,k) * binomial(2*n+1,2*k).

%F a(0) = 1, a(1) = 3 and n * (2*n+1) * (4*n-3) * a(n) = (4*n-1) * (4*n^2-2*n+1) * a(n-1) - 49 * (n-1) * (2*n-1) * (4*n+1) * a(n-2) for n > 1. - _Seiichi Manyama_, Aug 29 2020

%t a[n_] := Sum[(-3)^(n-k) * Binomial[2*k, k] * Binomial[2*n+1, 2*k], {k, 0, n}]; Array[a, 23, 0] (* _Amiram Eldar_, Apr 29 2021 *)

%o (PARI) N=40; x='x+O('x^N); Vec(sqrt(2/((1-2*x+49*x^2)*(1-7*x+sqrt(1-2*x+49*x^2)))))

%o (PARI) {a(n) = sum(k=0, n, (-3)^(n-k)*binomial(2*k, k)*binomial(2*n+1, 2*k))}

%Y Column k=3 of A337464.

%Y Cf. A245927, A337422.

%K sign

%O 0,2

%A _Seiichi Manyama_, Aug 28 2020

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)