|
%I #51 Jul 05 2020 23:36:19
%S 1,2,8,44,296,2312,20384,199376,2138336,24936416,314142848,4252773824,
%T 61594847360,950757812864,15586971531776,270569513970944,
%U 4959071121374720,95721139472072192,1941212789888952320,41271304403571227648,918030912312297752576,21325054720042613565440
%N Column 1 of triangle A111536.
%C Row sums of triangle in A200659. - _Philippe Deléham_, Nov 21 2011
%D A. N. Khovanskii. The Application of Continued Fractions and Their Generalizations to Problem in Approximation Theory. Groningen: Noordhoff, Netherlands, 1963. See p.141 (10.19).
%D H. P. Robinson, Letter to N. J. A. Sloane, Nov 19 1973.
%H Alois P. Heinz, <a href="/A111537/b111537.txt">Table of n, a(n) for n = 0..250</a>
%H Herman P. Robinson, <a href="/A003116/a003116_1.pdf">Letter to N. J. A. Sloane, Nov 19 1973</a>.
%F a(n) = A111536(n+1, 1) = 2*A111536(n, 0) = 2*A111529(n) for n >= 1.
%F G.f.: log(Sum_{n>=0} (n+1)!*x^n) = Sum_{n>=1} a(n)*x^n/n.
%F a(n+1) = (n+3)! - 2*(n+2)! - Sum_{k=0..n-1} (n-k+1)!*a(k+1).
%F a(n+1) is the moment of order n for the measure of density x*exp(-x)/((x*exp(-x)*Ei(x)-1)^2+(Pi*x*exp(-x))^2) on the interval 0..infinity.
%F G.f.: 1/(1-2*x/(1-2*x/(1-3*x/(1-3*x/(1-4*x/(1-4*x/(1-5*x/(1-...(continued fraction). - _Philippe Deléham_, Nov 21 2011
%F G.f. (1-U(0))/x; where U(k) = 1-x*(k+1)/(1-x*(k+2)/U(k+1)); (continued fraction, 2-step). - _Sergei N. Gladkovskii_, Jun 29 2012
%F G.f. -1 + 1/x + U(0)/x where U(k) = 2*x - 1 + 2*x*k - x^2*(k+1)*(k+2)/U(k+1), U(0)=x - W(1,1;-x)/W(1,2;-x), W(a,b,x)= 1 - a*b*x/1! + a*(a+1)*b*(b+1)*x^2/2! - ... + a*(a+1)*...*(a+n-1)*b*(b+1)*...*(b+n-1)*x^n/n! + ...; see [A. N. Khovanskii, p. 141 (10.19)]; (continued fraction, 1-step). - _Sergei N. Gladkovskii_, Aug 15 2012
%F G.f.: 1/Q(0), where Q(k) = 1 + k*x - x*(k+2)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 03 2013
%F G.f.: 1/x - 1/( x*G(0)), where G(k) = 1 - x*(k+1)/(x - 1/G(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Aug 03 2013
%F a(0) = 1; a(n) = n * a(n-1) + Sum_{k=0..n-1} a(k) * a(n-k-1). - _Ilya Gutkovskiy_, Jul 05 2020
%p a:= proc(n) option remember; `if`(n=0, 1,
%p n*(n+1)! -add((n-k+1)!*a(k), k=1..n-1))
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, May 06 2013
%t a[n_] := a[n] = If[n==0, 1, n*(n+1)! - Sum[(n-k+1)!*a[k], {k, 1, n-1}]]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 13 2017, after _Alois P. Heinz_ *)
%o (PARI) {a(n)=if(n<0,0,(matrix(n+2,n+2,m,j,if(m==j,1,if(m==j+1,-m+1, -(m-j-1)*polcoeff(log(sum(i=0,m,(i+1)!/1!*x^i)),m-j-1))))^-1)[n+2,2])}
%Y Cf. A111536, A111529, A200659.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Aug 06 2005
|
