# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a102840 Showing 1-1 of 1 %I A102840 #20 Jul 09 2024 09:02:24 %S A102840 0,1,3,0,-20,-45,21,308,540,-585,-4235,-5676,11232,54145,51975, %T A102840 -182400,-654160,-380205,2680425,7516400,1320900,-36753255,-82175665, %U A102840 24032700,477852900,850446025,-749925189,-5944471092,-8220606800,14049061455,71102953305,71989187536,-220682377872 %N A102840 a(0)=0, a(1)=1, a(n)=((2*n-1)*a(n-1)-5*n*a(n-2))/(n-1). %C A102840 n divides a(n) iff the binary representation of n ends with an even number of zeros (i.e. n is in A003159) %H A102840 Harvey P. Dale, Table of n, a(n) for n = 0..1000 %F A102840 log(abs(a(n))) is asymptotic to c*n where c=0.80... [c = log(5)/2 = 0.8047189562... - _Vaclav Kotesovec_, Feb 15 2019] %F A102840 a(n) ~ sqrt(n) * 5^(n/2) / sqrt(8*Pi) * ((sqrt(2 + sqrt(5)) + sqrt(38 + 25*sqrt(5)) / (16*n)) * sin(n*arctan(2)) - (sqrt(-2 + sqrt(5)) - sqrt(-38 + 25*sqrt(5)) / (16*n)) * cos(n*arctan(2))). - _Vaclav Kotesovec_, Feb 15 2019 %F A102840 From _Seiichi Manyama_, Jul 09 2024: (Start) %F A102840 G.f.: x/(1 - 2*x + 5*x^2)^(3/2). %F A102840 a(n+1) = binomial(n+2,2) * A343773(n). (End) %t A102840 RecurrenceTable[{-5 n a[n-2] + (2*n - 1) a[n-1] + (1 - n) a[n] == %t A102840 0, a[0] == 0, a[1] == 1}, a, {n, 0, 30}] (* _Vaclav Kotesovec_, Feb 15 2019 *) %t A102840 nxt[{n_,a_,b_}]:={n+1,b,(b(2n+1)-5a(n+1))/n}; NestList[nxt,{1,0,1},40][[;;,2]] (* _Harvey P. Dale_, Apr 22 2024 *) %o A102840 (PARI) a(n)=if(n<2,if(n,1,0),1/(n-1)*((2*n-1)*a(n-1)-5*n*a(n-2))) %Y A102840 Cf. A102839, A343773. %K A102840 sign %O A102840 0,3 %A A102840 _Benoit Cloitre_, Feb 27 2005 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE