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A163940 Triangle related to the divergent series 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ... for m >= -1. 13
1, 1, 0, 1, 2, 0, 1, 5, 3, 0, 1, 9, 17, 4, 0, 1, 14, 52, 49, 5, 0, 1, 20, 121, 246, 129, 6, 0, 1, 27, 240, 834, 1039, 321, 7, 0, 1, 35, 428, 2250, 5037, 4083, 769, 8, 0, 1, 44, 707, 5214, 18201, 27918, 15274, 1793, 9, 0, 1, 54, 1102, 10829, 54111, 133530, 145777, 55152, 4097, 10, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
The divergent series g(x,m) = Sum_{k >= 1} (-1)^(k+1)*k^m*k!*x^k, 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_{k = 0..m} a(m,k) * x^k, 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_{k = 0..m+1} S2(m+1, k) * x^k, m >= -1, lead to the Stirling numbers of the second kind, see A106800.
The formulas 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 formulas. 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.
LINKS
G. H. Hardy, Divergent Series, Oxford University Press, 1949. pp. 26-29 and pp. 7-8.
Maurice de Gosson, Branko Dragovich and Andrei Khrennikov, Some p-adic differential equations, (see Section 5), arxiv:math-ph/0010023, Oct 2000.
FORMULA
The generating functions of the right hand columns are Gf(p, x) = 1/((1 - (p-1)*x)^2 * Product_{k = 1..p-2} (1-k*x) ); Gf(1, x) = 1. For the first right hand column p = 1, for the second p = 2, etc..
From Peter Bala, Jul 23 2013: (Start)
Conjectural explicit formula: T(n,k) = Stirling2(n,n-k) + (n-k)*Sum_{j = 0..k-1} (-1)^j*Stirling2(n, n+1+j-k)*j! for 0 <= k <= n.
The n-th row polynomial R(n,x) appears to satisfy the recurrence equation R(n,x) = n*x^(n-1) + Sum_{k = 1..n-1} binomial(n,k+1)*x^(n-k-1)*R(k,x). The row polynomials appear to have only real zeros. (End)
EXAMPLE
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
MAPLE
nmax := 10; for p from 1 to nmax do Gf(p) := convert(series(1/((1-(p-1)*x)^2*product((1-k1*x), k1=1..p-2)), x, nmax+1-p), polynom); for q from 0 to nmax-p do a(p+q-1, q) := coeff(Gf(p), x, q) od: od: seq(seq(a(n, k), k=0..n), n=0..nmax-1);
# End program 1
nmax1:=nmax; A040027 := proc(n): if n = -1 then 0 elif n= 0 then 1 else add(binomial(n, k1-1)*A040027(n-k1), k1 = 1..n) fi: end: 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 nmax1 do g(1, n) := (-1)^n * (A040027(n) - A000110(n+1) * A073003) od;
# End program 2
MATHEMATICA
nmax = 11;
For[p = 1, p <= nmax, p++, gf = 1/((1-(p-1)*x)^2*Product[(1-k1*x), {k1, 1, p-2}]) + O[x]^(nmax-p+1) // Normal; For[q = 0, q <= nmax-p, q++, a[p+q-1, q] = Coefficient[gf, x, q]]];
Table[a[n, k], {n, 0, nmax-1}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 02 2019, from 1st Maple program *)
CROSSREFS
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), A014619.
Sequence in context: A293298 A079134 A175528 * A112340 A356264 A037186
KEYWORD
easy,nonn,tabl
AUTHOR
Johannes W. Meijer, Aug 13 2009
EXTENSIONS
Edited by Johannes W. Meijer, Sep 23 2012
STATUS
approved

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Last modified July 20 15:33 EDT 2024. Contains 374459 sequences. (Running on oeis4.)