A163940 -id:A163940 - cp's OEIS frontend

A073003 Decimal expansion of -exp(1)*Ei(-1), also called Gompertz's constant, or the Euler-Gompertz constant.

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

Offset: 0

Robert G. Wilson v, Aug 03 2002

0! - 1! + 2! - 3! + 4! - 5! + ... = (Borel) Sum_{n>=0} (-y)^n n! = KummerU(1,1,1/y)/y.
Decimal expansion of phi(1) where phi(x) = Integral_{t>=0} e^-t/(x+t) dt. - Benoit Cloitre, Apr 11 2003
The divergent series g(x=1,m) = 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ..., m => -1, is intimately related to Gompertz's constant. We discovered that g(x=1,m) = (-1)^m * (A040027(m) - A000110(m+1) * A073003) with A000110 the Bell numbers and A040027 a sequence that was published by Gould, see for more information A163940. - Johannes W. Meijer, Oct 16 2009
Named by Le Lionnais (1983) after the English self-educated mathematician and actuary Benjamin Gompertz (1779 - 1865). It was named the Euler-Gompertz constant by Finch (2003). Lagarias (2013) noted that he has not located this constant in Gompertz's writings. - Amiram Eldar, Aug 15 2020

			0.59634736232319407434107849936927937607417786015254878157348491...
With n := 10^5, Sum_{k >= 0} (n/(n + 1))^k/(n + k) = 0.5963(51...). - _Peter Bala_, Jun 19 2024

Bruce C. Berndt, Ramanujan's notebooks Part II, Springer, p. 171
Bruce C. Berndt, Ramanujan's notebooks Part I, Springer, p. 144-145.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 303, 424-425.
Francois Le Lionnais, Les nombres remarquables, Paris: Hermann, 1983. See p. 29.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 44, page 426.
H. S. Wall, Analytic Theory of Continued Fractions, Van Nostrand, New York, 1948, p. 356.

Robert Price, Table of n, a(n) for n = 0..10000
A. I. Aptekarev, On linear forms containing the Euler constant, arXiv:0902.1768 [math.NT], 2009. - _R. J. Mathar_, Feb 14 2009
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.
Xiangyu Ding and Lisa Hui Sun, Proof of Merca's stronger conjecture on truncated Jacobi triple product series, arXiv:2411.13818 [math.NT], 2024. See p. 20.
G. H. Hardy, Divergent Series, Oxford University Press, 1949. p. 29. - _Johannes W. Meijer_, Oct 16 2009
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., Vol. 50, No. 4 (2013), pp. 527-628, preprint, arXiv:1303.1856 [math.NT], 2013.
István Mezo, Gompertz constant, Gregory coefficients and a series of the logarithm function, Journal of Analysis & Number Theory, Vol. 2, No. 2 (2014), pp. 33-36.
Michael Penn, why some series are "regularizable", YouTube video (2023).
Simon Plouffe, -exp(1)*Ei(-1).
Tanguy Rivoal, On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant, Michigan Math. J., Vol. 61, No. 2 (2012), pp. 239-254.
Ed Sandifer, Divergent Series, How Euler Did It, MAA Online, June 2006. - _Johannes W. Meijer_, Oct 16 2009
Eric Weisstein's World of Mathematics, Gompertz Constant.
Eric Weisstein's World of Mathematics, Exponential Integral.

Cf. A000522 (arrangements), A001620, A000262, A002720, A002793, A058006 (alternating factorial sums), A091725, A099285, A153229, A201203, A245780, A283743 (Ei(1)/e), A321942, A369883.

SetDefaultRealField(RealField(100)); ExponentialIntegralE1(1)*Exp(1); // G. C. Greubel, Dec 04 2018

RealDigits[N[-Exp[1]*ExpIntegralEi[-1], 105]][[1]]
(* Second program: *)
G = 1/Fold[Function[2*#2 - #2^2/#1], 2, Reverse[Range[10^4]]] // N[#, 105]&; RealDigits[G] // First (* Jean-François Alcover, Sep 19 2014 *)

eint1(1)*exp(1) \\ Charles R Greathouse IV, Apr 23 2013

numerical_approx(exp_integral_e(1,1)*exp(1), digits=100) # G. C. Greubel, Dec 04 2018

phi(1) = e*(Sum_{k>=1} (-1)^(k-1)/(k*k!) - Gamma) = 0.596347362323194... where Gamma is the Euler constant.
G = 0.596347... = 1/(1+1/(1+1/(1+2/(1+2/(1+3/(1+3/(1+4/(1+4/(1+5/(1+5/(1+6/(... - Philippe Deléham, Aug 14 2005
Equals A001113*A099285. - Johannes W. Meijer, Oct 16 2009
From Peter Bala, Oct 11 2012: (Start)
Stieltjes found the continued fraction representation 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 numerators are in A002793 and the denominators in A002720.
Also, 1 - G has the continued fraction representation 1/(3 - 2/(5 - 6/(7 - ... -n*(n+1)/((2*n+3) - ...)))) with convergents beginning [1/3, 5/13, 29/73, 201/501, ...]. The numerators are in A201203 (unsigned) and the denominators are in A000262.
(End)
G = f(1) with f solution to the o.d.e. x^2*f'(x) + (x+1)*f(x)=1 such that f(0)=1. - Jean-François Alcover, May 28 2013
From Amiram Eldar, Aug 15 2020: (Start)
Equals Integral_{x=0..1} 1/(1-log(x)) dx.
Equals Integral_{x=1..oo} exp(1-x)/x dx.
Equals Integral_{x=0..oo} exp(-x)*log(x+1) dx.
Equals Integral_{x=0..oo} exp(-x)/(x+1) dx. (End)
From Gleb Koloskov, May 01 2021: (Start)
Equals Integral_{x=0..1} LambertW(e/x)-1 dx.
Equals Integral_{x=0..1} 1+1/LambertW(-1,-x/e) dx. (End)
Equals lim_{n->oo} A040027(n)/A000110(n+1). - Vaclav Kotesovec, Feb 22 2021
G = lim_{n->oo} A321942(n)/A000262(n). - Peter Bala, Mar 21 2022
Equals Sum_{n >= 1} 1/(n*L(n, -1)*L(n-1, -1)), where L(n, x) denotes the n-th Laguerre polynomial. This is the case x = 1 of the identity Integral_{t >= 0} exp(-t)/(x + t) dt = Sum_{n >= 1} 1/(n*L(n, -x)*L(n-1, -x)) valid for Re(x) > 0. - Peter Bala, Mar 21 2024
Equals lim_{n->oo} Sum_{k >= 0} (n/(n + 1))^k/(n + k). Cf. A099285. - Peter Bala, Jun 18 2024

Additional references from Gerald McGarvey, Oct 10 2005

Link corrected by Johannes W. Meijer, Aug 01 2009

A014619 Exponential generating function is -f(x) * Integral_{t = 0..x} exp(exp(-t) - 1) dt, where f(x) = exp(1 - x - exp(-x)) is the exponential generating function for A014182.

Original entry on oeis.org

-1, 1, 1, -5, 5, 21, -105, 141, 777, -5513, 13209, 39821, -527525, 2257425, -41511, -70561285, 531862173, -1559180499, -8858267353, 147780183829, -936560917615, 1352130196615, 38710924110081, -487251979381019, 2846575686392251, 872653153712201

Offset: 1

Noam D. Elkies

sign

Branko Dragovich, On Summation of p-Adic Series, arXiv:1702.02569 [math.NT], 2017.
Branko Dragovich, Andrei Yu. Khrennikov, and Natasa Z. Misic, Summation of p-Adic Functional Series in Integer Points, arXiv:1508.05079 [math.NT], 2015.
B. Dragovich and N. Z. Misic, p-Adic invariant summation of some p-adic functional series, P-Adic Numbers, Ultrametric Analysis, and Applications, October 2014, Volume 6, Issue 4, pp 275-283.

Cf. A000670, A040027, A163940.

a[n_] := Sum[(-1)^(n - k + 1) * StirlingS2[n + 1, k + 1] * ((-1)^k * k! * Subfactorial[-k - 1] - Subfactorial[-1]), {k, 0, n}]; Table[a[n] // FullSimplify, {n, 1, 26}] (* Jean-François Alcover, Jan 09 2014, after Vladeta Jovovic *)
nmax = 25; Rest[CoefficientList[Series[E^(-E^(-x) - x) * (Gamma[0, -1] - Gamma[0, -E^(-x)]), {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, May 10 2024 *)

a(n)=local(A,B);if(n<0,0, A=exp(-x+x*O(x^n)); B=exp(A-1);n!*polcoeff(-intformal(B)*A/B,n))

E.g.f. A(x) = y satisfies y'' + y'(2-exp(-x)) + y = 0. - Michael Somos, Mar 11 2004
a(n) = Sum_{k = 0..n} (-1)^(n-k+1)*Stirling2(n+1, k+1)*A003422(k). - Vladeta Jovovic, Jan 06 2005
The sequence b(n) = (-1)^n*a(n) satisfies the recurrence: b(n) = -Sum_{i = 1..n} b(i-1)*C(n, i), b(0) = -1. - Ralf Stephan, Feb 24 2005
From Peter Bala, Mar 23 2024: (Start)
It appears that a(n) = Sum_{k = 1..n+1} binomial(n+1, k)*a(k). See Dragovich 2017, Table 1.
If true then the following hold: setting a(0) = -1 then
a(n) = Sum_{k = 1..n-1} (-1)^(n-k)*binomial(n-1, k-1)*a(k-1);
the o.g.f F(x) = - ( x/(1 + x)^2  + x^2/((1 + x)*(1 + 2*x)^2) + x^3/((1 + x)*(1 + 2*x)*(1 + 3*x)^2) + ... ) - Cf. A040027;
F(x) = - x/(1 + x)^2 + x/(1 + x)^2*F(x/(1 + x)). (End)

More terms from Jason Earls, Jun 28 2001

A163972 The MC polynomials.

Original entry on oeis.org

1, 0, 3, 1, 0, 2, 45, 22, 3, 0, 0, 10, 107, 61, 13, 1, 0, -48, 20, 2100, 14855, 9168, 2390, 300, 15, 0, 0, -336, 92, 6320, 33765, 21803, 6378, 1010, 85, 3, 0, 11520, -2016, -198296, 33012, 2199246, 9547461, 6331782, 1994265, 362474, 39375, 2394, 63

Offset: 1

Johannes W. Meijer, Aug 13 2009

The a(n,p) polynomials, see below with the extra p for the column number, generate the coefficients of the left hand columns of triangle A163940. These polynomials are interesting in their own right. They have many curious properties; e.g., for p >= 1: a(n=1, p) = p, a(n=0, p) = 0, a(n = -1, p) = (-1)^(p+1), a(n=-2,p) = (-1)^(p+1)*(2)^(p-2) and a(n = -(2*p+1), 2*p) = 0, which is the outermost zero of the a(n, 2*p); for p >= 10: a(n=-10, p) = -362880*10^(p-10); etc.
The numbers in the denominators of the a(n,p) are the Minkowski numbers A053657.
The Maple program generates the coefficients of the polynomials that appear in the numerators of the a(n,p), see the sequence above. We have made use of a nice little program that Peter Luschny recently wrote for the Minkowski numbers! For the an(p,k) in the Maple program for p >= 1 we have 0 <= k <= (2*p-2). A word of caution: The value of nmax has to be chosen sufficiently large in order to let Maple find the o.g.f.s.
The zero patterns of the a(n,p) polynomials resemble the Montezuma Cypress (Taxodium mucronatum). A famous Montezuma Cypress is 'El Arbol del Tule' (the Tule tree) in Mexico. It is the second stoutest tree in the world, circumference 36 meters, and is approximately 1500 years old. Considering this I propose to call the a(n,p) polynomials the MC polynomials.
The row sums equal n*A053657(n). [Johannes W. Meijer, Nov 29 2012]

			The a(n,p) formulas of the first few left hand columns of the A163940 triangle (p is the column number):
a(n,1) = (1)/1
a(n,2) = (0 + 3*n + n^2)/2
a(n,3) = (0 + 2*n + 45*n^2+ 22*n^3 + 3*n^4)/24
a(n,4) = (0 + 0*n + 10*n^2 + 107*n^3 + 61*n^4 + 13*n^5 + n^6)/48
a(n,5) = (0 - 48*n + 20*n^2 + 2100*n^3 + 14855*n^4 + 9168*n^5 + 2390*n^6 + 300*n^7 + 15*n^8)/5760
a(n,6) = (0 + 0*n -336*n^2 +92*n^3 +6320*n^4 +33765*n^5 +21803*n^6 +6378*n^7 +1010*n^8 +85*n^9 +3*n^10)/11520
a(n,7) = (0 + 11520*n -2016*n^2 -198296*n^3 +33012*n^4 +2199246*n^5 +9547461*n^6+ 6331782*n^7 +1994265*n^8 +362474*n^9 +39375*n^10 +2394*n^11 +63*n^12)/2903040

Johannes W. Meijer, The zeros of the MC polynomials, pdf and jpg.

A000012, A000096, A163943 and A163944 are the first four left hand columns of A163940.

Cf. A053657 (Minkowski), A163402 and A075264.

pmax:=6; nmax:=70; with(genfunc): A053657 := proc(n) local P, p, q, s, r; P := select(isprime, [$2..n]); r:=1; for p in P do s := 0: q := p-1; do if q > (n-1) then break fi; s := s + iquo(n-1, q); q := q*p; od; r := r * p^s; od; r end: for px from 1 to nmax do Gf(px):= convert(series(1/((1-(px-1)*x)^2*product((1-k*x), k=1..px-2)),x,nmax+1-px),polynom): for qy from 0 to nmax-px do a(px+qy,qy):=coeff(Gf(px),x,qy) od; od: for p from 1 to pmax do f(x):=0: for ny from p to nmax do f(x):=f(x)+a(ny,p-1)*x^(ny-p) od: f(x):= series(f(x),x, nmax): Gx:=convert(%, ratpoly): rgf_sequence('recur',Gx,x,G,n): a(n,p):=sort(simplify (rgf_expand(Gx,x,n)),n): f(p):=sort(a(n,p)*A053657(p),n,ascending): for k from 0 to 2*p-2 do an(p,k):= coeff(f(p),n,k) od; od: T:=1: for p from 1 to pmax do for k from 0 to 2*p-2 do a(T):=an(p,k): T:=T+1 od: od: seq(a(n),n=1..T-1); for p from 1 to pmax do seq(an(p,k),k=0..2*p-2) od; for p from 1 to pmax do MC(n,p):=sort(a(n,p),n,ascending) od;

cp's OEIS Frontend

A073003 Decimal expansion of -exp(1)*Ei(-1), also called Gompertz's constant, or the Euler-Gompertz constant.

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A014619 Exponential generating function is -f(x) * Integral_{t = 0..x} exp(exp(-t) - 1) dt, where f(x) = exp(1 - x - exp(-x)) is the exponential generating function for A014182.

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A163972 The MC polynomials.

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