A073003 -id:A073003 - cp's OEIS frontend

A153229 a(0) = 0, a(1) = 1, and for n >= 2, a(n) = (n-1) * a(n-2) + (n-2) * a(n-1).

0, 1, 0, 2, 4, 20, 100, 620, 4420, 35900, 326980, 3301820, 36614980, 442386620, 5784634180, 81393657020, 1226280710980, 19696509177020, 335990918918980, 6066382786809020, 115578717622022980, 2317323290554617020, 48773618881154822980, 1075227108896452857020

Offset: 0

Shaojun Ying (dolphinysj(AT)gmail.com), Dec 21 2008

Previous name was: Weighted Fibonacci numbers.
From Peter Bala, Aug 18 2013: (Start)
The sequence occurs in the evaluation of the integral I(n) := Integral_{u >= 0} exp(-u)*u^n/(1 + u) du.
The result is I(n) = A153229(n) + (-1)^n*I(0), where I(0) = Integral_{u >= 0} exp(-u)/(1 + u) du = 0.5963473623... is known as Gompertz's constant. See A073003.
Note also that I(n) = n!*Integral_{u >= 0} exp(-u)/(1 + u)^(n+1) du. (End)
((-1)^(n+1))*a(n) = p(n,-1), where the polynomials p are defined at A248664. - Clark Kimberling, Oct 11 2014

			a(20) = 19 * a(18) + 18 * a(19) = 19 * 335990918918980 + 18 * 6066382786809020 = 6383827459460620 + 109194890162562360 = 115578717622022980

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000045, A000255, A000587, A058006.
First differences of A136580.
Column k=0 of A303697 (for n>0).

unsigned long a(unsigned int n) {
if (n == 0) return 0;
if (n == 1) return 1;
return (n - 1) * a(n - 2) + (n - 2) * a(n - 1); }

t1 := sum(n!*x^n, n=0..100): F := series(t1/(1+x), x, 100): for i from 0 to 40 do printf(`%d, `, i!-coeff(F, x, i)) od: # Zerinvary Lajos, Mar 22 2009
# second Maple program:
a:= proc(n) a(n):= `if`(n<2, n, (n-1)*a(n-2) +(n-2)*a(n-1)) end:
seq(a(n), n=0..25); # Alois P. Heinz, May 24 2013

Join[{a = 0}, Table[b = n! - a; a = b, {n, 0, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[n]==(n-1)a[n-2]+(n-2)a[n-1]},a,{n,30}] (* Harvey P. Dale, May 01 2020 *)

a(n)=if(n,my(t=(-1)^n);-t-sum(i=1,n-1,t*=-i),0); \\ Charles R Greathouse IV, Jun 28 2011

a(0) = 0, a(1) = 1, and for n >= 2, a(n) = (n-1) * a(n-2) + (n-2) * a(n-1).
For n>=1, a(n) = A058006(n-1) * (-1)^(n-1).
G.f.: G(0)*x/(1+x)/2, where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
G.f.: 2*x/(1+x)/G(0), where G(k)= 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 29 2013
G.f.: W(0)*x/(1+sqrt(x))/(1+x), where W(k) = 1 + sqrt(x)/( 1 - sqrt(x)*(k+1)/(sqrt(x)*(k+1) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 17 2013
a(n) ~ (n-1)! * (1 - 1/n + 1/n^3 + 1/n^4 - 2/n^5 - 9/n^6 - 9/n^7 + 50/n^8 + 267/n^9 + 413/n^10), where numerators are Rao Uppuluri-Carpenter numbers, see A000587. - Vaclav Kotesovec, Mar 16 2015
E.g.f.: exp(1)/exp(x)*(Ei(1, 1-x)-Ei(1, 1)). - Alois P. Heinz, Jul 05 2018
a(n) = Sum_{k = 0..n-1} (-1)^(n-k-1) * k!. - Peter Bala, Dec 05 2024

Edited by Max Alekseyev, Jul 05 2010

Better name by Joerg Arndt, Aug 17 2013

A201203 Alternating row sums of triangle A201201: first associated monic Laguerre-Sonin(e) polynomials with parameter alpha=1 evaluated at x=-1.

Original entry on oeis.org

1, -5, 29, -201, 1631, -15173, 159093, -1854893, 23788271, -332613321, 5033396573, -81929955953, 1426898945343, -26468817431501, 520884561854501, -10836674357638293, 237603001692915983, -5475288709200573713, 132276033079845108621

Offset: 0

Wolfdieter Lang, Dec 06 2011

G. C. Greubel, Table of n, a(n) for n = 0..440
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017, Table 2 on p. 13.

Cf. A201201, A201202 (row sums), A073003, A002793.

A201203 := proc(n)
    add((-1)^k*A201201(n,k),k=0..n) ;
end proc:
seq(A201203(n),n=0..20) ; # R. J. Mathar, Dec 07 2011

Flatten[{1,RecurrenceTable[{n*(1+n)*a[-2+n]+(3+2*n)*a[-1+n] +a[n]==0, a[1]==-5,a[2]==29}, a, {n, 20}]}] (* Vaclav Kotesovec, Oct 19 2013 *)

a(n) = Sum_{k=0..n} ((-1)^k)*A201201(n,k), n>=0.
a(n)+(2*n+3)*a(n-1)+n*(n+1)*a(n-2)=0, a(-1)=0, a(0)=1. - R. J. Mathar, Dec 07 2011
From Wolfdieter Lang, Dec 11 2011: (Start)
E.g.f. from A201201 with x=-1, z->x: g(x) = exp(1/(1+x))*(3+2*x)*(exp(-1) + (Ei(1,1/(1+x))-Ei(1,1)))/(1+x)^4-(2+x)/(1+x)^3, with the exponential integral Ei.
This e.g.f. satisfies the homogeneous ordinary second-order differential equation (1+x)^2*(d^2(g(x))/dx^2) + (7+6*x)*(d(g(x))/dx)+6*g(x), with g(0)=1 and (d(g(x))/dx){x=0} = -5. This is equivalent to the recurrence conjectured above by _R. J. Mathar, which is thus proved.
(End)
Let G denote Gompertz's constant A073003. The unsigned sequence is the sequence of numerators in the convergents coming from the infinite continued fraction expansion 1 - G = 1/(3 - 2/(5 - 6/(7 - ... - n*(n+1)/((2*n+3) - ...)))). The sequence of convergents begins [1/3, 5/13, 29/73, 201/501, ...]. The denominators are in A000262. - Peter Bala, Aug 19 2013
a(n) ~ (-1)^n * 2^(-1/2)*(exp(-1)-Ei(1,1)) * exp(2*sqrt(n)-n+1/2) * n^(n+7/4) * (1+91/(48*sqrt(n))), where Ei(1,1) = 0.21938393439552... = G / exp(1), where G = 0.596347362323194... is the Gompertz constant (see A073003). - Vaclav Kotesovec, Oct 19 2013

R. J. Mathar conjecture corrected and proved by Wolfdieter Lang, Dec 11 2011

A007549 Number of increasing rooted connected graphs where every block is a complete graph.

Original entry on oeis.org

1, 1, 3, 14, 89, 716, 6967, 79524, 1041541, 15393100, 253377811, 4596600004, 91112351537, 1959073928124, 45414287553455, 1129046241331316, 29965290866974493, 845605519848379436, 25282324544244718411, 798348403914242674980, 26549922456617388029641

Offset: 1

N. J. A. Sloane

In an increasing rooted graph, nodes are numbered and the numbers increase as you move away from the root.

(a(n+1)/a(n))/n tends to 1/A073003 = 1.676875... (same limit as A029768). - Vaclav Kotesovec, Jul 26 2014

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 1..410 (first 200 terms from Vincenzo Librandi)
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Cf. A007563, A030019, A035051-A035053.
Cf. A029768.
Row sums of A078341. Column k=1 of A264436.

exptr:= proc(p) local g; g:= proc(n) option remember; p(n) +add(binomial(n-1, k-1) *p(k) *g(n-k), k=1..n-1) end: end: b:= exptr(exptr(a)): a:= n-> `if`(n=0, 1, b(n-1)): seq(a(n), n=1..30); # Alois P. Heinz, Oct 07 2008

exptr[p_] := Module[{g}, g[n_] := g[n] = p[n] + Sum[ Binomial[n-1, k-1]*p[k]*g[n-k], {k, 1, n-1}]; g]; b = exptr[ exptr[a] ]; a[n_] := If[n == 0, 1, b[n-1]]; Table[ a[n], {n, 1, 19}] (* Jean-François Alcover, May 10 2012, after Alois P. Heinz *)

Shifts left when exponentiated twice.
Conjecture: a(n) = Sum_{i=0..2^(n-2) - 1} b(i) for n > 1 with a(1) = 1 where b(n) = (L(n) + 2)*b(f(n)) + Sum_{k=0..L(n) - 1} (1 - R(n,k))*b(f(n) + 2^k*(1 - R(n,k))) for n > 0 with b(0) = 1, L(n) = A000523(n), f(n) = A053645(n) and where R(n,k) = floor(n/2^k) mod 2. Here R(n,k) is the (k+1)-th bit from the right side in the binary expansion of n. - Mikhail Kurkov, Jul 21 2024
Conjecture: a(n) = D^(n-1)(exp(x)) evaluated at x = 0, where D denotes the operator exp(x)*(1 + x)*d/dx. - Peter Bala, Feb 24 2025

New description from Christian G. Bower, Oct 15 1998

A002793 a(n) = 2na(n-1) - (n-1)^2a(n-2).

Original entry on oeis.org

0, 1, 4, 20, 124, 920, 7940, 78040, 859580, 10477880, 139931620, 2030707640, 31805257340, 534514790680, 9591325648580, 182974870484120, 3697147584561340, 78861451031150840, 1770536585183202980, 41729280102868841080, 1030007496863617367420, 26568602827124392999640

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

nonn

From Wolfdieter Lang, Dec 12 2011: (Start)
r(n) = a(n+1)*(-1)^n, n >= 0, gives the alternating row sums of the coefficient triangle A199577, i.e., r(n)=La_n(1;0,-1), with the monic first associated Laguerre polynomials with parameter alpha=0 evaluated at x=-1.
The e.g.f. for these row sums r(n) is g(x) = -(2+x)*exp(1/(1+x))*(Ei(1,1/(1+x))-Ei(1,1))/(1+x)^3 + 1/(1+x)^2, with the exponential integral Ei(1,x) = Gamma(0,x).
This e.g.f. satisfies the homogeneous ordinary second-order differential equation (1+x)^2*(d^2/dx^2)g(x) + (6+5*x)*(d/dx)g(x) + 4*g(x) = 0, g(0)=1, (d/dx)g(x)|_{x=0}=-4.
This e.g.f. g(x) is equivalent to the recurrence
  b(n)= -2*(n+1)*b(n-1) - n^2*b(n-2), b(-1)=0, b(0)=1.
Therefore, the e.g.f. of a(n) is A(x)=int(g(-x),x), with A(0)=0. This agrees with the e.g.f. given below in the formula section by Max Alekseyev.
(End)

J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 78.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 356.

G. C. Greubel, Table of n, a(n) for n = 0..440
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages)

Bisection of A056952. A199577 (alternating row sums, unsigned).

Cf. A002720, A073003.

I:=[1, 4]; [0] cat [n le 2 select I[n] else 2*n*Self(n-1) - (n-1)^2*Self(n-2): n in [1..30]]; // G. C. Greubel, May 16 2018

Flatten[{0,RecurrenceTable[{(-1+n)^2 a[-2+n]-2 n a[-1+n]+a[n]==0,a[1]==1,a[2]==4}, a, {n, 20}]}] (* Vaclav Kotesovec, Oct 19 2013 *)
nxt[{n_,a_,b_}]:={n+1,b,2(n+1)b-n^2 a}; NestList[nxt,{1,0,1},30][[All,2]] (* Harvey P. Dale, Sep 06 2022 *)

A058006(n) = sum(k=0,n, (-1)^k*k! );
a(n) = if (n<=1, n, sum(k=1, n, (k+1) * A058006(k-1) * binomial(n,k) * (n-1)! / (k-1)! ) ); /* Joerg Arndt, Oct 12 2012 */

{a(n)=if(n==1,1,polcoeff(1-sum(m=1, n-1, a(m)*x^m*(1-(m+1)*x+x*O(x^n))^2), n))} \\ Paul D. Hanna, Feb 06 2013

From Max Alekseyev, Jul 06 2010: (Start)
For n > 1, a(n) = Sum_{k=1..n} (k+1) * A058006(k-1) * binomial(n,k) * (n-1)! / (k-1)!.
E.g.f.: (Gamma(0,1) - Gamma(0,1/(1-x))) * exp(1/(1-x)) / (1-x). (End)
From Peter Bala, Oct 11 2012: (Start)
Numerators in the sequence of convergents of Stieltjes's continued fraction for A073003, the Euler-Gompertz constant G := int {x = 0..oo} 1/(1+x)*exp(-x) dx:
G = 1/(2 - 1^2/(4 - 2^2/(6 - 3^2/(8 - ...)))). See [Wall, Chapter 18, (92.7) with a = 1]. The sequence of convergents to the continued fraction begins [1/2, 4/7, 20/34, 124/209, ...]. The denominators are in A002720.
(End)
G.f.: x = Sum_{n>=1} a(n) * x^n * (1 - (n+1)*x)^2. - Paul D. Hanna, Feb 06 2013
a(n) ~ G * exp(2*sqrt(n) - n - 1/2) * n^(n+1/4) / sqrt(2) * (1 + 31/(48*sqrt(n))), where G = 0.596347362323194... is the Gompertz constant (see A073003). - Vaclav Kotesovec, Oct 19 2013

Edited by Max Alekseyev, Jul 13 2010

A239069 Decimal expansion of gamma - Ei(-1).

Original entry on oeis.org

7, 9, 6, 5, 9, 9, 5, 9, 9, 2, 9, 7, 0, 5, 3, 1, 3, 4, 2, 8, 3, 6, 7, 5, 8, 6, 5, 5, 4, 2, 5, 2, 4, 0, 8, 0, 0, 7, 3, 2, 0, 6, 6, 2, 9, 3, 4, 6, 8, 3, 1, 8, 0, 6, 3, 8, 3, 7, 4, 5, 8, 4, 7, 9, 5, 8, 4, 3, 6, 4, 2, 5, 3, 3, 6, 8, 0, 6, 2, 1, 5, 6, 5, 9, 1, 5, 7, 3, 1, 4, 3, 2, 6, 8, 8, 3, 9, 9, 9, 4

Offset: 0

Jonathan Sondow, Mar 12 2014

See crossrefs sequences for other comments, references, links, and formulas.

			0.7965995992970531342836758655425240800732066293468...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 37, table 37:7:1 at page 355.

G. C. Greubel, Table of n, a(n) for n = 0..10000
J. C. Lagarias, Euler's constant: Euler's work and modern developments, arXiv:1303.1856 [math.NT], 2013-2014; Bull. Amer. Math. Soc., 50 (2013), 527-628; see p. 553.

Cf. A001008, A001113, A001563, A001620, A002805, A073003, A099285.

RealDigits[EulerGamma - ExpIntegralEi[-1], 10, 100][[1]]

Euler + eint1(1,1)[1] \\ Michel Marcus, Aug 01 2020

Equals (the Euler-Mascheroni constant) - (the exponential integral at -1) = A001620 + A099285.
Equals (the Euler-Mascheroni constant) + (Gompertz's constant / e) = A001620 + (A073003 / A001113).
Equals Sum_{n>=1} (-1)^(n-1) / A001563(n) = Sum_{n>=1} (-1)^(n-1) / (n*n!).
Equals -Integral_{x=0..1} log(x)/exp(x) dx. - Amiram Eldar, Aug 01 2020
Equals (1/e) * Sum_{k>=1} H(k)/k!, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jun 25 2021

A347952 Decimal expansion of exp(1) * (gamma - Ei(-1)).

Original entry on oeis.org

2, 1, 6, 5, 3, 8, 2, 2, 1, 5, 3, 2, 6, 9, 3, 6, 3, 5, 9, 4, 2, 0, 9, 8, 6, 3, 4, 8, 4, 9, 2, 4, 3, 0, 5, 6, 8, 3, 8, 1, 4, 2, 0, 7, 6, 7, 7, 4, 1, 4, 4, 3, 6, 9, 0, 2, 3, 0, 1, 3, 9, 1, 7, 1, 8, 9, 4, 9, 4, 2, 4, 2, 5, 7, 9, 7, 7, 9, 8, 7, 1, 7, 9, 7, 6, 9, 2, 6, 0, 3, 5, 1, 4, 1, 5, 5, 6, 7, 5, 7, 2, 6, 7, 6, 4, 7, 5, 3, 4, 8

Offset: 1

Ilya Gutkovskiy, Oct 23 2021

			2.16538221532693635942098634849243056838142076774144369...

Cf. A001008, A001113, A001620, A002805, A073003, A099285, A239069, A244274, A348573.

RealDigits[Exp[1] (EulerGamma - ExpIntegralEi[-1]), 10, 110] [[1]]

exp(1)*(Euler + eint1(1)) \\ Michel Marcus, Oct 24 2021

Equals Sum_{k>=1} H(k) / k!, where H(k) is the k-th harmonic number.

Equals -Integral_{x=0..1} exp(x)*log(1-x) dx. - Amiram Eldar, Oct 23 2021

A283743 Decimal expansion of Ei(1)/e, where Ei is the exponential integral function.

Original entry on oeis.org

6, 9, 7, 1, 7, 4, 8, 8, 3, 2, 3, 5, 0, 6, 6, 0, 6, 8, 7, 6, 5, 4, 7, 8, 6, 8, 1, 9, 1, 9, 5, 5, 1, 5, 9, 5, 3, 1, 7, 1, 7, 5, 4, 3, 0, 9, 5, 4, 3, 6, 9, 5, 1, 7, 3, 2, 0, 0, 5, 4, 8, 0, 7, 7, 8, 9, 4, 5, 4, 1, 1, 5, 1, 9, 5, 1, 4, 4, 2, 6, 9, 6, 2, 9, 1, 0, 0, 5, 3, 0, 3, 0, 3, 3, 3, 9, 1, 1, 4, 0, 0, 6

Offset: 0

Jean-François Alcover, Mar 15 2017

Can be considered the value of the divergent series -0! - 1! - 2! - ... ; see Lagarias reference Section 2.5. - Harry Richman, Jun 14 2020.

			0.6971748832350660687654786819195515953171754309543695173200548...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 44, equation 44:5:10 at page 426.

Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, arXiv:1303.1856 [math.NT], 2013; Bull. Amer. Math. Soc., 50 (2013), 527-628.
Eric Weisstein's World of Mathematics, Exponential Integral.
Eric Weisstein's World of Mathematics, Subfactorial.

Cf. A000166 (subfactorials), A061382 (Pi/e, the imaginary part of subfactorial(-1)), A091725 (Ei(1)), A073003 (-exp(1)*Ei(-1)).

RealDigits[ExpIntegralEi[1]/E, 10, 102][[1]]

real(-eint1(-1)/exp(1)) \\ Michel Marcus, Jun 15 2020

Equals Re(subfactorial(-1)) = Re(Gamma(0,-1)/e).

Equals Sum_{k=1..oo} (-1)^k*psi(k)/Gamma(k), where psi denotes the digamma function (see Spanier and Oldham). - Stefano Spezia, Jan 04 2025

A245780 Decimal expansion of (1-C_2)/e, a constant connected with two-sided generalized Fibonacci sequences, where C_2 is the Euler-Gompertz constant.

Original entry on oeis.org

1, 4, 8, 4, 9, 5, 5, 0, 6, 7, 7, 5, 9, 2, 2, 0, 4, 7, 9, 1, 8, 3, 5, 9, 9, 9, 4, 7, 0, 1, 3, 3, 9, 2, 1, 8, 4, 1, 4, 7, 6, 3, 8, 3, 7, 6, 2, 4, 8, 5, 9, 6, 2, 6, 9, 2, 9, 8, 5, 8, 1, 8, 8, 6, 2, 3, 8, 9, 2, 7, 9, 7, 1, 8, 5, 7, 5, 8, 2, 5, 8, 6, 3, 4, 9, 3, 7, 0, 2, 3, 3, 1, 0, 7, 8, 2, 3, 9, 3, 7, 9

Offset: 0

Jean-François Alcover, Aug 01 2014

			0.148495506775922047918359994701339218414763837624859626929858...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.2 Euler-Gompertz Constant, p. 426.

Walther Janous, Problem 1552, Crux Mathematicorum, Vol. 16, No. 6 (1990), p. 171; Solution to Problem 1552, by Richard Katz, ibid., Vol. 17, No. 7 (1991), pp. 223-224.
Peter Fishburn, Andrew Odlyzko and Fred Roberts, Two-sided generalized Fibonacci sequences, The Fibonacci Quarterly, Vol. 27, No. 4 (1989), pp. 352-361.

Cf. A073003 (C_2), A099285 (C_2 / e).

$RecursionLimit = 10^4; digits = 101; m0 = 100; dm = 100; Clear[g]; g[m_] := g[m] = (Clear[a, b, f]; b[n_] := 2*n; a[n_ /; n >= m] = 0; a[1] = 1; a[2] = -1; a[n_] := -(n-1)^2; f[m] = b[m]; f[n_] := f[n] = b[n] + a[n+1]/f[n+1]; (1 - f[0])/E); g[m0]; g[m = m0 + dm]; While[RealDigits[g[m], 10, digits] != RealDigits[g[m - dm], 10, digits], m = m + dm]; RealDigits[g[m], 10, digits] // First
(* or, as verification: *) RealDigits[1/E + ExpIntegralEi[-1], 10, digits] // First

1/exp(1) - eint1(1,1)[1] \\ Michel Marcus, Aug 06 2020

Equals 1/e + Ei(-1), where Ei is the exponential integral function.
Equals Integral_{x=0..1} exp(-1/x) dx. - Amiram Eldar, Aug 06 2020
Equals Integral_{x=1..+oo} exp(-x)/x^2 dx. - Jianing Song, Oct 03 2021
Equals lim_{n->oo} (Sum_{k=1..n-1} (k/(k+1))^n)/n (Janous, 1990). - Amiram Eldar, Apr 03 2022

A249385 Decimal expansion of gamma - 2*Ei(-1), one of the Tauberian constants, where Ei is the exponential integral function.

Original entry on oeis.org

1, 0, 1, 5, 9, 8, 3, 5, 3, 3, 6, 9, 2, 5, 7, 3, 4, 0, 7, 9, 6, 0, 8, 3, 9, 6, 4, 1, 0, 0, 2, 6, 4, 5, 7, 2, 9, 1, 0, 4, 2, 5, 3, 9, 2, 2, 7, 5, 3, 7, 4, 0, 0, 1, 3, 9, 6, 1, 7, 2, 4, 4, 6, 1, 0, 3, 2, 0, 0, 5, 1, 2, 3, 8, 9, 5, 9, 4, 7, 7, 6, 0, 3, 8, 1, 3, 6, 7, 5, 6, 5, 3, 6, 2, 0, 2, 1, 2, 4, 9, 4, 2, 4

Offset: 1

Jean-François Alcover, Oct 27 2014

			1.01598353369257340796083964100264572910425392275374...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Tauberian Constants, August 30, 2004 [Cached copy, with permission of the author]
Eric Weisstein's MathWorld, Exponential Integral

Cf. A001620, A073003, A099285, A248472.

evalf(gamma - 2*Ei(-1), 120); # Vaclav Kotesovec, Oct 27 2014

RealDigits[ EulerGamma - 2*ExpIntegralEi[-1], 10, 103] // First

default(realprecision, 100); Euler + 2*eint1(1) \\ G. C. Greubel, Sep 04 2018

Also equals gamma + 2*G/e, where G is the Euler-Gompertz constant 0.596347...

Equals A001620 + 2*A073003/e. - G. C. Greubel, Sep 04 2018

A321942 A sequence related to the Euler-Gompertz constant.

Original entry on oeis.org

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

Offset: 1

Richard P. Brent, Dec 12 2018

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.

			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.

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.

Cf. A000262, A073003, A201203.

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

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 *)

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

cp's OEIS Frontend

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A002793 a(n) = 2n*a(n-1) - (n-1)^2*a(n-2).

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