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A077230
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Numerators of coefficients of series expansion of a certain integral in the theory of charged particle beams.
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2
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2, 1, -7, 5, -787, 763, -893209, 2885597, -1153151299, 261937547, -3997632829, 30141297349, -4101190700056349, 2948796705108299, -320676905674696783, 43360062621189833, -5848606947453449297743, 1963629536423819469923, -575654781675816234791672323
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OFFSET
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0,1
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COMMENTS
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The integral is Integrate[1/Sqrt[Log[y]],{y,1,x}]=Sqrt[Pi]*Erfi[Sqrt[Log[x]] with series expansion Sqrt[x-1]*Sum[c(i)*(x-1)^(i-1),{i,0,19}].
Numerator(c(n)) = A077230(n) (this sequence); denominator(c(n)) = A077231(n).
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REFERENCES
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M. Reiser, Theory and design of charged particle beams. J. Wiley, N.Y. 1994.
S. Humphries, Charged particle beams. J. Wiley, N.Y. 1990.
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LINKS
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FORMULA
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a(n) = numerator(sum(k=1..n, 4^(-k)*binomial(2*k,k)*sum(j=1..k, (j!*binomial(k,j)*(-1)^(j)*stirling1(n+j,j))/(n+j)!))), n>0, a(0)=2. - Vladimir Kruchinin, Feb 18 2015
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EXAMPLE
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Series expansion is Sqrt[x-1]*(2 + 1/6 (x-1) -7/240 (x-1)^2+ 5/448 (x-1)^3 -...), hence a(0)=2, a(1)=1, a(2)=-7, a(3)=5, etc.
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MATHEMATICA
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a[n_] := If [n == 0, 2, Sum[4^(-k)*Binomial[2*k, k]*Sum[(j!*Binomial[k, j]*(-1)^j* StirlingS1[n+j, j])/(n+j)!, {j, 1, k}], {k, 1, n}]] // Numerator; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 18 2015, after Vladimir Kruchinin *)
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PROG
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(Maxima)
a(n):=if n=0 then 2 else num(sum(4^(-k)*binomial(2*k, k)*sum((j!*binomial(k, j)*(-1)^(j)*stirling1(n+j, j))/(n+j)!, j, 1, k), k, 1, n)); /* Vladimir Kruchinin, Feb 18 2015 */
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CROSSREFS
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KEYWORD
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sign,frac
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AUTHOR
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STATUS
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approved
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