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A357719 Expansion of e.g.f. cos( 2 * log(1+x) ). 3

%I #10 Oct 12 2022 08:58:32

%S 1,0,-4,12,-28,40,200,-3360,35680,-357120,3644800,-38896000,437756800,

%T -5206406400,65372153600,-864339840000,11991424640000,

%U -173800340480000,2617640829440000,-40693929269760000,647089190924800000,-10383194262604800000

%N Expansion of e.g.f. cos( 2 * log(1+x) ).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.

%F a(n) = Sum_{k=0..floor(n/2)} (-4)^k * Stirling1(n,2*k).

%F a(n) = (-1)^n * ( (2 * i)_n + (-2 * i)_n )/2, where (x)_n is the Pochhammer symbol and i is the imaginary unit.

%F a(0) = 1, a(1) = 0; a(n) = -(2*n-3) * a(n-1) - (n^2-4*n+8) * a(n-2).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(cos(2*log(1+x))))

%o (PARI) a(n) = sum(k=0, n\2, (-4)^k*stirling(n, 2*k, 1));

%o (PARI) a(n) = (-1)^n*(prod(k=0, n-1, 2*I+k)+prod(k=0, n-1, -2*I+k))/2;

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; v[2]=0; for(i=2, n, v[i+1]=-(2*i-3)*v[i]-(i^2-4*i+8)*v[i-1]); v;

%Y Column k=4 of A357720.

%Y Cf. A357711, A357727.

%K sign

%O 0,3

%A _Seiichi Manyama_, Oct 10 2022

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Last modified June 23 13:32 EDT 2024. Contains 373648 sequences. (Running on oeis4.)