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%I #13 Nov 27 2020 10:47:01
%S 1,13,121,18581,305071,61203943,4353296221,6669149100757,
%T 206772189255571,128970681211645873,24697503335329725121,
%U 45583359018138184284551,6235055851689626935206871,7982707567621372702411448803,2955418704408380517540605162821,40101878131071637461151318174173269
%N a(n) = (-1)^(n + 1)*3^(2*n + 1)*Euler(2*n + 1, 1/3)*2^(valuation_{2}(2*(n + 1))), the Steinhaus-Euler sequence S_{3}(n).
%H Sandor Csörgö, Gordon Simons, <a href="http://www.math.uni.wroc.pl/~pms/files/14.2/Abstract/14.2.1.abs.pdf">On Steinhaus' resolution of the St. Petersburg paradox</a>, Probab. Math. Statist. 14 (1993), 157-172. MR1321758 (96b:60017).
%e The array of the general case S_{k}(n) starts:
%e [k]
%e [1] -1, -1, -1, -17, -31, -691, -5461, ... [-A002425]
%e [2] 0, 0, 0, 0, 0, 0, 0, ...
%e [3] 1, 13, 121, 18581, 305071, 61203943, 4353296221, ... [this seq.]
%e [4] 2, 44, 722, 196888, 5746082, 2049374444, 259141449842, ...
%e [5] 3, 99, 2523, 1074243, 48982293, 27296351769, 5393115879063, ...
%e ...
%p GenEuler := k -> (n -> (-1)^n*(-k)^(2*n+1)*euler(2*n+1, 1/k)):
%p Steinhaus := n -> 2^padic[ordp](2*(n+1), 2):
%p seq(Steinhaus(n)*GenEuler(3)(n), n = 0..15);
%t GenEuler[n_, k_] := (-1)^n (-k)^(2 n + 1) EulerE[2 n + 1, 1/k] ;
%t Steinhaus[n_] := 2^IntegerExponent[2*(n+1), 2];
%t a[n_] := GenEuler[n, 3] Steinhaus[n]; Table[a[n], {n, 0, 15}]
%Y Cf. A002425, A339058.
%K nonn
%O 0,2
%A _Peter Luschny_, Nov 27 2020
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