The On-Line Encyclopedia of Integer Sequences (OEIS)

A077230 revision #15

A077230

Numerators of coefficients of series expansion of a certain integral in the theory of charged particle beams.

2, 1, -7, 5, -787, 763, -893209, 2885597, -1153151299, 261937547, -3997632829, 30141297349, -4101190700056349, 2948796705108299, -320676905674696783, 43360062621189833, -5848606947453449297743, 1963629536423819469923, -575654781675816234791672323

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OFFSET

0,1

COMMENTS

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).

REFERENCES

M. Reiser, Theory and design of charged particle beams. J. Wiley, N.Y. 1994.

S. Humphries, Charged particle beams. J. Wiley, N.Y. 1990.

LINKS

Table of n, a(n) for n=0..18.

FORMULA

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

a(n) = numerator(Sum_{k=0..n} binomial(1/2-n, k)*binomial(2*n-1/2, n-k)*Stirling2(n+k, k)*k!/((1-2*n)*(n+k)!)), n>0, a(0)=2. - Tani Akinari, Nov 05 2024

EXAMPLE

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.

MATHEMATICA

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 *)

PROG

(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 */

(Maxima) a(n):=if n=0 then 2 else num(sum(binomial(1/2-n, k)*binomial(2*n-1/2, n-k)*stirling2(n+k, k)*k!/((1-2*n)*(n+k)!), k, 0, n)); /* Tani Akinari, Nov 05 2024 */

CROSSREFS

Cf. A077231.

Sequence in context: A342747 A365320 A356732 * A244238 A019668 A091700

Adjacent sequences: A077227 A077228 A077229 * A077231 A077232 A077233

KEYWORD

sign,frac

AUTHOR

Zak Seidov, Oct 31 2002

STATUS

editing