A331778 -id:A331778 - cp's OEIS frontend

A331777 Numerators of coefficients in asymptotic expansion of exp(2*(H_k-gamma))/k^2 in powers of 1/k, where H_k are the harmonic numbers A001008/A002805 and gamma is the Euler-Mascheroni constant A001620.

1, 1, 1, 0, -1, 1, -1, -43, 1831, 949, -137309, -85511, 3404045159, 777985057, -21024051077, -2192231411, 467347169033357, 10187765700589, -11741590582705819219, -3086703970985605357, 169597995722575162268081, 19606186988235984155519, -62715098968866173387571821

Offset: 0

N. J. A. Sloane, Feb 09 2020

Chao-Ping Chen, On the coefficients of asymptotic expansion for the harmonic number by Ramanujan, The Ramanujan Journal, 40:2 (2016), 279-290. See Eq. (2.11).

Denominators are in A331778.

Cf. A001008, A001620, A002805, A093334, A238813, A331943.

Numerator[CoefficientList[Series[Exp[2*(HarmonicNumber[k] - EulerGamma)]/k^2, {k, Infinity, 25}], 1/k]] (* Vaclav Kotesovec, Feb 10 2020 *)

Sign of a(7) corrected and more terms from Vaclav Kotesovec, Feb 10 2020

A331943 a(n) = n^2 + 1 - ceiling((n + 2)/3).

Original entry on oeis.org

1, 3, 8, 15, 23, 34, 47, 61, 78, 97, 117, 140, 165, 191, 220, 251, 283, 318, 355, 393, 434, 477, 521, 568, 617, 667, 720, 775, 831, 890, 951, 1013, 1078, 1145, 1213, 1284, 1357, 1431, 1508, 1587, 1667, 1750, 1835, 1921, 2010, 2101, 2193, 2288, 2385, 2483, 2584

Offset: 1

Hugo Pfoertner, Feb 10 2020

Related to expansion of exp(2*(H_k-gamma))/k^2 in powers of 1/k as given by A331777/A331778.

The agreement with the results of the PARI code needs an explanation. All numerators corresponding to the computed denominators are 1.

Hugo Pfoertner, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A001008, A001620, A002805, A008620, A331777, A331778.

Table[n^2+1-Ceiling[(n+2)/3],{n,60}] (* or *) LinearRecurrence[{2,-1,1,-2,1},{1,3,8,15,23},60] (* Harvey P. Dale, Aug 30 2021 *)

H(n)=sum(j=1,n,1/j);
A(k)=exp(2*(H(k)-Euler))/k^2;
for(k=1,51,r=(1/k)*(A(k)-1);print1(denominator(bestappr(r,k*k)),", "))

From Colin Barker, Feb 10 2020: (Start)
G.f.: x*(1 + x + 3*x^2 + x^3) / ((1 - x)^3*(1 + x + x^2)).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>5.
(End)
E.g.f.: (1/9)*(3*exp(x)*x*(2 + 3*x) + 2*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2)). - Stefano Spezia, Feb 14 2020

cp's OEIS Frontend

A331777 Numerators of coefficients in asymptotic expansion of exp(2*(H_k-gamma))/k^2 in powers of 1/k, where H_k are the harmonic numbers A001008/A002805 and gamma is the Euler-Mascheroni constant A001620.

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