A321942 A sequence related to the Euler-Gompertz constant.

1, 2, 8, 44, 300, 2420, 22460, 235260, 2741660, 35152820, 491459820, 7436765660, 121046445260, 2108118579060, 39104985755420, 769549656815420, 16009942093608060, 351030466622487860, 8089084984387984460, 195421894806240545820, 4938445392988428283820

Offset

1,2

Comments

a(n) satisfies the recurrence a(n) = (2n-1)*a(n-1) - (n-1)*(n-2)*a(n-2) for n > 2, with initial conditions a(1)=1, a(2)=2.

The same recurrence is satisfied by A000262(n), but with different initial conditions.

The limit of a(n)/A000262(n) as n tends to infinity is the Euler-Gompertz constant G = e*E1(1), where E1 is an exponential integral. The decimal representation of G is given by A073003.

The convergents of the c.f. G = 1-1/(3-1*2/(5-2*3/(7-3*4/(9-...)))) are (a(n)/A000262(n)) = (1, 2/3, 8/13, 44/73, ...). The c.f. is equivalent to Bala's c.f. for 1-G given in the entry for A073003.

a(n)/A000262(n) - G ~ 2*Pi*exp(1-4*sqrt(n)) as n tends to infinity.

a(n)/n! ~ G*exp(2*sqrt(n))/(2*n^(3/4)*sqrt(Pi*e)) as n tends to infinity.

a(n) = A000262(n) - |A201203(n-2)| for n >= 2.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..444

Richard P. Brent, M. L. Glasser, and Anthony J. Guttmann, A Conjectured Integer Sequence Arising From the Exponential Integral, arXiv:1812.00316 [math.NT], 2018.

NIST Digital Library of Mathematical Functions, Exponential, Logarithmic, Sine and Cosine Integrals.

Formula

a(n) = (2n-1)*a(n-1) - (n-1)*(n-2)*a(n-2) for n > 2.

E.g.f.: exp(x/(1-x))*(G - E1(x/(1-x))), where G is the Euler-Gompertz constant and E1 is an exponential integral.

Conjecture: Integral_{x = 0..oo} (x/(1 + x))^n*exp(-x) dx = 1/(n-1)!*( a(n) - A000262(n)*G ), where G = Integral_{x = 0..oo} exp(-x)/(1 + x) dx is the Euler-Gompertz constant A073003. - Peter Bala, Mar 20 2022

Example

a(3) = (2*3-1)*a(2) - 2*1*a(1) = 5*2 - 2*1 = 8.
a(3) = A000262(3) - |A201203(1)| = 13 - |5| = 8.

Maple

a:= proc(n) option remember; `if`(n<3, n,
      (2*n-1)*a(n-1) -(n-1)*(n-2)*a(n-2))
    end:
seq(a(n), n=1..23);  # Alois P. Heinz, Dec 12 2018

Mathematica

a[n_] := a[n] = (2n-1)a[n-1] - (n-1)(n-2)a[n-2]; a[1] = 1; a[2] = 2;
Array[a, 21] (* Jean-François Alcover, Oct 06 2019 *)

Crossrefs

Cf. A000262, A073003, A201203.

Sequence in context: A051045 A112912 A303613 * A124467 A075792 A052897

Adjacent sequences: A321939 A321940 A321941 * A321943 A321944 A321945

Keyword

nonn,easy

Author

Richard P. Brent, Dec 12 2018

Status

approved