A077230 - OEIS

%I #14 Feb 18 2015 12:43:23

%S 2,1,-7,5,-787,763,-893209,2885597,-1153151299,261937547,-3997632829,

%T 30141297349,-4101190700056349,2948796705108299,-320676905674696783,

%U 43360062621189833,-5848606947453449297743,1963629536423819469923,-575654781675816234791672323

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

%C 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}].

%C Numerator(c(n)) = A077230(n) (this sequence); denominator(c(n)) = A077231(n).

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

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

%F 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

%e 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.

%t 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_ *)

%o (Maxima)

%o 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 */

%Y Cf. A077231.

%K sign,frac

%O 0,1

%A _Zak Seidov_, Oct 31 2002