0,5

The divergent series g(x,m) = sum((-1)^(k+1)*k^m*k!*x^k, k= 1..infinity), m=>-1, are related to the higher order exponential integrals E(x,m,n=1), see A163931.

Hardy, see the link below, describes how Euler came to the rather surprising conclusion that g(x,-1) = exp(1/x)*Ei(1,1/x) with Ei(1,x) = E(x,m=1,n=1). From this result it follows inmediately that g(x,0) = 1 - g(x,-1). Following in Euler's footsteps we discovered that g(x,m) = (-1)^(m) * (M(x,m)*x - ST(x,m)* Ei(1,1/x) * exp(1/x))/x^(m+1), m =>-1.

So g(x=1,m) = (-1)^m*(A040027(m) - A000110 (m+1)*A073003), with A040027(m = -1) = 0. We observe that A073003 = - exp(1)*Ei(-1) is Gompertz's constant, A000110 are the Bell numbers and A040027 was published a few years ago by Gould.

The polynomial coefficients of the M(x,m) = sum(a(m,k) * x^k, k = 0..m), for m =>0 lead to the triangle given above. We point out that M(x,m=-1) = 0.

The polynomial coefficients of the ST(x,m) = sum(S2(m+1, k) * x^k, k = 0..m+1), m =>-1, lead to the Stirling numbers of the second kind, see A106800.

The formulae that generate the coefficients of the left hand columns lead to the Minkowski numbers A053657. We have a closer look at them in A163972.

The right hand columns have simple generating functions, see the formulae. We used them in the first Maple program to generate the triangle coefficients (n >= 0 and 0 <= k <= n). The second Maple program calculates the values of g(x,m) for m=>-1, at x=1.

G.H. Hardy, Divergent Series, Oxford University Press, 1949. pp. 26-29 and pp. 7-8.

The generating functions of the right hand columns are Gf(p) = 1/((1-(p-1)*x)^2*product((1-k*x), k=1..p-2)); Gf(1) = 1. For the first right hand column p=1, for the second p=2, etc..

The first few triangle rows are:

[1]

[1, 0]

[1, 2, 0]

[1, 5, 3, 0]

[1, 9, 17, 4, 0]

[1, 14, 52, 49, 5, 0]

The first few M(x,m) are:

M(x,m=0) = 1

M(x,m=1) = 1 + 0*x

M(x,m=2) = 1 + 2*x + 0*x^2

M(x,m=3) = 1 + 5*x + 3*x^2 + 0*x^3

The first few ST(x,m) are:

ST(x,m=-1) = 1

ST(x,m=0) = 1 + 0*x

ST(x,m=1) = 1 + 1*x + 0*x^2

ST(x,m=2) = 1 + 3*x + x^2 + 0*x^3

ST(x,m=3) = 1 + 6*x + 7*x^2 + x^3 + 0*x^4

The first few g(x,m) are:

g(x,-1) = (-1)*(- (1)*Ei(1,1/x)*exp(1/x))/x^0

g(x,0) = (1)*((1)*x - (1)*Ei(1,1/x)*exp(1/x))/x^1

g(x,1) = (-1)*((1)*x - (1+ x)*Ei(1,1/x)*exp(1/x))/x^2

g(x,2) = (1)*((1+2*x)*x - (1+3*x+x^2)*Ei(1,1/x)*exp(1/x))/x^3

g(x,3) = (-1)*((1+5*x+3*x^2)*x - (1+6*x+7*x^2+x^3)*Ei(1,1/x)*exp(1/x))/x^4

restart; nmax:=11; imax:=nmax: for p from 1 to imax do Gf(p):=convert(series(1/((1-(p-1)*x)^2*product((1-k*x), k=1..p-2)), x, imax+1-p), polynom); for q from 0 to imax-p do a(p+q-1, q):=coeff(Gf(p), x, q) od: od: T:=0: for n from 0 to nmax-1 do for k from 0 to n do a(T):=a(n, k); T:=T+1; od: od: seq(a(n), n=0..T-1);

restart; nmax:=11; A040027(-1):=0: A040027(0):=1: for n from 1 to nmax do A040027(n) := sum(binomial(n, k-1)*A040027(n-k), k = 1..n) od: A000110 := proc(n) option remember; if n <= 1 then 1 else add( binomial(n-1, i)*A000110(n-1-i), i=0..n-1); fi; end: A073003 := - exp(1) * Ei(-1): for n from -1 to nmax do g(1, n):= (-1)^n*(A040027(n)-A000110(n+1)*A073003) od;

The row sums equal A040027 (Gould).

A000007, A000027, A000337, A163941 and A163942 are the first five right hand columns.

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

Cf. A163931, A163972, A106800 (Stirling2), A000110 (Bell), A073003 (Gompertz), A053657 (Minkowski).

Sequence in context: A125183 A092583 A079134 this_sequence A112340 A037186 A004483

Adjacent sequences: A163937 A163938 A163939 this_sequence A163941 A163942 A163943

easy,nonn,tabl

Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 13 2009