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A163940
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Triangle related to the divergent series 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ... for m >= -1.
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12
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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
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OFFSET
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0,5
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COMMENTS
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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.
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 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.
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LINKS
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G. H. Hardy, Divergent Series, Oxford University Press, 1949. pp. 26-29 and pp. 7-8.
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FORMULA
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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..
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)
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EXAMPLE
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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The row sums equal A040027 (Gould).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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