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

Author

Vladimir Shevelev, Sep 22 2017

Keywords

• nonn
• frac

Comments

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,... .

References

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

Links

• 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.

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

PARI

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

Python

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

Formula

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

Extensions

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