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

%I A357703 #23 Jul 23 2025 09:08:02
%S A357703 1,0,3,9,42,240,1614,12474,108900,1059696,11371932,133410420,
%T A357703 1698541416,23324023008,343606235544,5405580540360,90445832210448,
%U A357703 1603781918563968,30042007763367600,592788643008571152,12289695299276133024,267079782474700715520
%N A357703 Expansion of e.g.f. cosh( sqrt(3) * log(1-x) ).
%H A357703 Seiichi Manyama, <a href="/A357703/b357703.txt">Table of n, a(n) for n = 0..449</a>
%H A357703 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%F A357703 a(n) = Sum_{k=0..floor(n/2)} 3^k * |Stirling1(n,2*k)|.
%F A357703 a(n) = ( (sqrt(3))_n + (-sqrt(3))_n )/2, where (x)_n is the Pochhammer symbol.
%F A357703 a(0) = 1, a(1) = 0; a(n) = (2*n-3) * a(n-1) - (n^2-4*n+1) * a(n-2).
%o A357703 (PARI) my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cosh(sqrt(3)*log(1-x)))))
%o A357703 (PARI) a(n) = sum(k=0, n\2, 3^k*abs(stirling(n, 2*k, 1)));
%o A357703 (PARI) a(n) = round((prod(k=0, n-1, sqrt(3)+k)+prod(k=0, n-1, -sqrt(3)+k)))/2;
%o A357703 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=2, n, v[i+1]=(2*i-3)*v[i]-(i^2-4*i+1)*v[i-1]); v;
%Y A357703 Column k=3 of A357712.
%Y A357703 Cf. A357615.
%K A357703 nonn
%O A357703 0,3
%A A357703 _Seiichi Manyama_, Oct 10 2022