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 A357703 Expansion of e.g.f. cosh( sqrt(3) * log(1-x) ). 3

%I #19 Oct 10 2022 14:51:54

%S 1,0,3,9,42,240,1614,12474,108900,1059696,11371932,133410420,

%T 1698541416,23324023008,343606235544,5405580540360,90445832210448,

%U 1603781918563968,30042007763367600,592788643008571152,12289695299276133024,267079782474700715520

%N Expansion of e.g.f. cosh( sqrt(3) * log(1-x) ).

%H Seiichi Manyama, <a href="/A357703/b357703.txt">Table of n, a(n) for n = 0..449</a>

%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) = ( (sqrt(3))_n + (-sqrt(3))_n )/2, where (x)_n is the Pochhammer symbol.

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

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

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

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

%Y Column k=3 of A357712.

%Y Cf. A357615.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Oct 10 2022

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Last modified July 15 09:15 EDT 2024. Contains 374324 sequences. (Running on oeis4.)