A157805 -id:A157805 - cp's OEIS frontend

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

1, 9, 125, 32977, 971919, 358472059, 47622059953, 137818710619425, 8141400285401267, 9740358918723188381, 3597069206174040366021, 12859671622917809034800123, 3419734700063005545155284375, 8538628250545609672426471056711, 6181704419438256867205044161777369

Offset: 1

Vladimir Shevelev, Sep 22 2017

Conjecture: a(n) is divisible by (2*n-1)^2.
Robert G. Wilson v verified this conjecture up to 5000.
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,... .

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 1972, Ch. 23.

Robert G. Wilson v, Table of n, a(n) for n = 1..215
Vladimir Shevelev, On a Luschny question, arXiv:1708.08096 [math.NT], 2017.

Cf. A002425, A006519, A007814, A157805, A292706.

A291897 := n -> euler(2*n-1, n)*2^(padic[ordp](2*n, 2)):
seq(A291897(n), n=1..15); # Peter Luschny, Sep 22 2017

f[n_] := Numerator@ EulerE[2 n - 1, n]; Array[f, 15] (* Robert G. Wilson v, Sep 22 2017 *)
Table[2^IntegerExponent[2n, 2] EulerE[2 n-1, n], {n,1,15}] (* Peter Luschny, Sep 22 2017 *)

a(n) = numerator(subst(eulerpol(2*n-1, 'x), 'x, n)); \\ Michel Marcus, Sep 21 2021

from sympy import euler
def A291897(n): return euler((n<<1)-1,n).p # Chai Wah Wu, Jul 07 2022

a(n) = (E(2*n-1,n) + (-1)^(n-1)*E(2*n-1,0))*A006519(2*n) + A002425(n).
a(n) = 2*(-1)^n*A292706(n)*A006519(2*n) + A002425(n).
a(n) = E(2*n-1, n)*2^A007814(2*n). - Peter Luschny, Sep 22 2017

More terms from Peter J. C. Moses, Sep 22 2017

A143074 Numerator of Euler(n,2).

Original entry on oeis.org

1, 3, 2, 9, 2, 3, 2, 33, 2, -27, 2, 699, 2, -5457, 2, 929601, 2, -3202287, 2, 221930589, 2, -4722116517, 2, 968383680843, 2, -14717667114147, 2, 2093660879252679, 2, -86125672563201177, 2, 129848163681107302017, 2, -868320396104950823607, 2, 209390615747646519456969

Offset: 0

N. J. A. Sloane, Nov 10 2009

			By the formula, we have a(1) = 2*2 - 1 = 3, a(3) = 2*4 + 1 = 9, a(5) = 2*2 - 1 = 3, a(7) = 2*8 + 17 = 33, a(9) = 2*2 - 31 = -27, etc. - _Vladimir Shevelev_, Sep 04 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vladimir Shevelev, On a Luschny question, arXiv:1708.08096 [math.NT], 2017.
Vladimir Shevelev, A formula for numerator of Euler(n,k), Wed Sep 06 2017.

For denominators see A006519.

Cf. A002425, A157805, A157808, A157812, A157827, A157835, A157856, A157864, A157875, A157886, A157907.

Numerator[EulerE[Range[0,40],2]] (* Vincenzo Librandi, May 03 2012 *)

For even n, a(n) = 2 - delta(n,0), where delta is the Kronecker symbol;
for n==1 (mod 4), a(n) = 2*A006519(n+1) - A002425((n+1)/2);
for n==3 (mod 4), a(n) = 2*A006519(n+1) + A002425((n+1)/2). - Vladimir Shevelev, Sep 04 2017

A292706 a(n) = 1/2((-1)^nE(2n-1,n) - E(2n-1,0)), where E(n,x) is the Euler polynomial.

Original entry on oeis.org

0, 1, -31, 2060, -242972, 44808921, -11905513623, 4306834677808, -2035350070549744, 1217544864812657225, -899267301542329562375, 803729476432302540694956, -854933675015747706872042556, 1067328531318200947345698975505, -1545426104859564195269842899644047

Offset: 1

Vladimir Shevelev, Sep 21 2017

sign

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 1972, Ch. 23.

Cf. A143074, A157805.

Table[((-1)^n EulerE[2n-1,n]-EulerE[2n-1,0])/2,{n,10}]
Map[Total[(Map[(-1)^# (#-1)&,Range[#]])^(2#-1)]&,Range[10]]
(* Peter J. C. Moses, Sep 21 2017 *)

a(n) = sum(k=1, n-1, (-1)^(k+1)*k^(2*n-1)); \\ Michel Marcus, Sep 22 2017

a(n) = 1^(2*n-1) - 2^(2*n-1) + ... + (-1)^n*(n-1)^(2*n-1).

|a(n)| ~ 1/(1+e^(-2))*(n-1)^(2*n-1) = 0.88079707...*(n-1)^(2*n-1) as n goes to infinity.

More terms from Peter J. C. Moses, Sep 21 2017

cp's OEIS Frontend

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

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A143074 Numerator of Euler(n,2).

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A292706 a(n) = 1/2((-1)^nE(2n-1,n) - E(2n-1,0)), where E(n,x) is the Euler polynomial.

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cp's OEIS Frontend

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

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Maple

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A143074 Numerator of Euler(n,2).

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Programs

Mathematica

Formula

A292706 a(n) = 1/2*((-1)^n*E(2*n-1,n) - E(2*n-1,0)), where E(n,x) is the Euler polynomial.

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Formula

Extensions

A292706 a(n) = 1/2((-1)^nE(2n-1,n) - E(2n-1,0)), where E(n,x) is the Euler polynomial.