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A291897 Numerator of E(2*n-1,n), where E(n,x) is the Euler polynomial.

%I A291897 #48 Jul 08 2022 08:21:00
%S A291897 1,9,125,32977,971919,358472059,47622059953,137818710619425,
%T A291897 8141400285401267,9740358918723188381,3597069206174040366021,
%U A291897 12859671622917809034800123,3419734700063005545155284375,8538628250545609672426471056711,6181704419438256867205044161777369
%N A291897 Numerator of E(2*n-1,n), where E(n,x) is the Euler polynomial.
%C A291897 Conjecture: a(n) is divisible by (2*n-1)^2.
%C A291897 _Robert G. Wilson v_ verified this conjecture up to 5000.
%C A291897 Note that sometimes a(n) is divisible by (2n-1)^3, for example, for n = 1,3,7,9,... when 2*n-1 = 1,5,13,17,... .
%D A291897 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 1972, Ch. 23.
%H A291897 Robert G. Wilson v, <a href="/A291897/b291897.txt">Table of n, a(n) for n = 1..215</a>
%H A291897 Vladimir Shevelev, <a href="https://arxiv.org/abs/1708.08096">On a Luschny question</a>, arXiv:1708.08096 [math.NT], 2017.
%F A291897 a(n) = (E(2*n-1,n) + (-1)^(n-1)*E(2*n-1,0))*A006519(2*n) + A002425(n).
%F A291897 a(n) = 2*(-1)^n*A292706(n)*A006519(2*n) + A002425(n).
%F A291897 a(n) = E(2*n-1, n)*2^A007814(2*n). - _Peter Luschny_, Sep 22 2017
%p A291897 A291897 := n -> euler(2*n-1, n)*2^(padic[ordp](2*n, 2)):
%p A291897 seq(A291897(n), n=1..15); # _Peter Luschny_, Sep 22 2017
%t A291897 f[n_] := Numerator@ EulerE[2 n - 1, n]; Array[f, 15] (* _Robert G. Wilson v_, Sep 22 2017 *)
%t A291897 Table[2^IntegerExponent[2n, 2] EulerE[2 n-1, n], {n,1,15}] (* _Peter Luschny_, Sep 22 2017 *)
%o A291897 (PARI) a(n) = numerator(subst(eulerpol(2*n-1, 'x), 'x, n)); \\ _Michel Marcus_, Sep 21 2021
%o A291897 (Python)
%o A291897 from sympy import euler
%o A291897 def A291897(n): return euler((n<<1)-1,n).p # _Chai Wah Wu_, Jul 07 2022
%Y A291897 Cf. A002425, A006519, A007814, A157805, A292706.
%K A291897 nonn,frac
%O A291897 1,2
%A A291897 _Vladimir Shevelev_, Sep 22 2017
%E A291897 More terms from _Peter J. C. Moses_, Sep 22 2017