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 A357718 Expansion of e.g.f. cos( sqrt(3) * log(1+x) ). 2

%I #11 Oct 12 2022 08:58:37

%S 1,0,-3,9,-24,60,-84,-756,13104,-157248,1795248,-20900880,254007936,

%T -3250473408,43922668608,-626830626240,9437477107968,-149644407564288,

%U 2493958878657792,-43592393744250624,797394015216175104,-15230735270523601920

%N Expansion of e.g.f. cos( sqrt(3) * 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)} (-3)^k * Stirling1(n,2*k).

%F a(n) = (-1)^n * ( (sqrt(3) * i)_n + (-sqrt(3) * 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+7) * a(n-2).

%o (PARI) my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cos(sqrt(3)*log(1+x)))))

%o (PARI) a(n) = sum(k=0, n\2, (-3)^k*stirling(n, 2*k, 1));

%o (PARI) a(n) = (-1)^n*round((prod(k=0, n-1, sqrt(3)*I+k)+prod(k=0, n-1, -sqrt(3)*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+7)*v[i-1]); v;

%Y Column k=3 of A357720.

%Y Cf. A357703, A357726.

%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.)