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The utility of large networks, particularly social networks, with n participants is given by the terms a(n) of this sequence. This assertion is known as Reed’s Law, see the Wikipedia link. - Johannes W. Meijer, Jun 03 2019

Wikipedia, <a href="https://en.wikipedia.org/wiki/Reed%27s_law">Reed’s Law</a>

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D. C. Gijswijt, <a href="httphttps://www.pyth.eu/jaargangenuploads/user/ArchiefPDF/Pyth55-3.pdf">Krulgetallen</a>, Pythagoras, 55ste Jaargang, Nummer 3, Jan 2016, (Article in Dutch about this sequence, see pages 10-13, cover and back cover).

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We define the higher-order exponential integrals by E(x,m,n) = x^(n-1)*int(E(t,m-1,n)/t^n, t=x..infinity) for m => 0 =1 and n => =1 with E(x,m=0,n) = exp(-x), see Meijer and Baken.

E(x=1,m=2,n=1) = gamma^2/2 + Pi^2/12 + Sum_{k=1..infinity} ((-1)^k/(k^2*k!)).

E(x=0,n,m) = (1/(n-1))^m for n=>=2.

E(x,m,n) = (-1)^m * ((-x)^(n-1)/(n-1)!) *sum Sum_{kz=0..floor(m/2)}(alpha (kz, n)*(G(m-2*kz, n) ) + (-1) ^m * ((-x)^(n-1)/(n-1)!) * Sum_{kz=0..floor(m/2)} (Sum_{i=1..m-2*kz}(alpha (kz, n) *G(m-2*kz-i, n)*logln(x)^i/i!,i=)) + (-1..)^m-2 *kz))) + Sum_{ kx=0..(n-2)} ((-x)^kx/((kx-n+1)^m*kx!)) + (-1)^m * Sum_{ky=n..infinity}((-x)^ky /(( ky-n+1)^m*ky!))).

Johannes W. Meijer & and _Nico Baken (n.h.g.baken(AT)tudelft.nl), _, Aug 13 2009, Aug 17 2009

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The higher order exponential integrals, see A163931, are defined by E(x,m,n) = x^(n-1)*int(E(t,m-1,n)/t^n, t=x..infinity) for m =>0 =1 and n>=1, with E(x,m=0,n) = exp(-x).

G(k,n) = (1/k)*((gamma-sum(p^*G(k-1,n)) - (1/k),*Sum_{p=1..n-1}(p^(-1))* G(k-1,n) +sum (1/k)* Sum_{i=0..k-2}(Zeta(k-i) -sum(p^* G(i,n)) - (1/k-)*Sum_{i)),=0..k-2}(Sum_{p=1..n-1}(p^(i-k)) * G(i,n),i=0..k-2)) with G(0,n) = 1 for k => =0 and n => =1.

GF(z,n) = GAMMA(n-z)/GAMMA(n).

(gamma - G(1,n)) equals A001008(n-1)/A002805(n-1) for n =>=2.

The structure of the G(k,n=1) formula leads formulas lead (replace gamma by G and Zeta by Z) to A036039.

Johannes W. Meijer & and _Nico Baken (n.h.g.baken(AT)tudelft.nl), _, Aug 13 2009, Aug 17 2009

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The higher order exponential integrals, see A163931, are defined by E(x,m,n) = x^(n-1)*int(E(t,m-1,n)/t^n, t=x..infinity) for m >=0 1 and n>=1, with E(x,m=0,n) = exp(-x).

The first Maple program uses the alpha(k,n) formula and the second the GF(z,n) to generate the alpha(k,n) coefficients in each column.

alpha(k,n) = (1/k)*sum(sum(p^(-2 *( Sum_{i=0..k-i)),1} (Sum_{p = 0..n-1}(p^(2*i-2*k))*alpha(i, n), i = 0..k-1) with alpha(0,n) = 1, k >= 0 and n >= 1.

GF(z,n) = product((1-(z/k)^2)^(-1), k = 1..n-1) = (Pi*z/sin(Pi*z))/(Beta(n+z,n-z)/Beta(n,n))

coln:=4; nmax1 := nmax+1: 16; for n from 0 to nmax1 do A008955(n, 0):=1 end do: for n from 0 to nmax1 do A008955(n, n) := (n!)^2 end do: for n from 1 to nmax1 do for m from 1 to n-1 do A008955(n, m) := A008955(n-1, m-1)*n^2 + A008955(n-1, m) end do: end do: m:=coln-1: f(m):=0: for n from 0 to m do f(m) := f(m) + (-1)^(n + m)*A008955(m, n)*z^(2*m-2*n) od: GF(z, coln) := m!^2/f(m): GF(z, coln):=series(GF(z, coln), z, nmaxnmax1);

Cf. A163931 (E(x, m, n)), A163930 (gamma(k, n)).

The GF(z,n) lead to A008955.

The denominators of a(1,n), n >= 2, lead to A007407.

Johannes W. Meijer & and _Nico Baken (n.h.g.baken(AT)tudelft.nl), _, Aug 13 2009, Aug 17 2009

Edited by Johannes W. Meijer, Sep 22 2012

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