A162478 Expansion of 1/sqrt(1-4x/(1-x)^4).
1, 2, 14, 88, 566, 3722, 24856, 167868, 1143462, 7841434, 54065574, 374437404, 2602879712, 18150990238, 126918338116, 889551010728, 6247598686710, 43958881741086, 309801915943318, 2186512103767096, 15452093394793006
Offset: 0
Links
- Robert Israel, Table of n, a(n) for n = 0..1163
Programs
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Maple
f:= gfun:-rectoproc({n*a(n) +(7-9*n)*a(n-1) +2*(7*n-17)*a(n-2) +10*(3-n)*a(n-3) +5*(n-4)*a(n-4) +(5-n)*a(n-5),a(0)=1,a(1)=2,a(2)=14,a(3)=88,a(4)=566},a(n),remember): map(f, [$0..50]); # Robert Israel, Dec 27 2017 -
Mathematica
CoefficientList[Series[1/Sqrt[1-(4x)/(1-x)^4],{x,0,20}],x] (* Harvey P. Dale, Aug 02 2016 *)
Formula
G.f.: 1/(1-2x/((1-x)^4-x/(1-x/((1-x)^4-x/(1-... (continued fraction);
a(n) = Sum_{k=0..n} C(n+3k-1,n-k)*A000984(k).
D-finite with recurrence: n*a(n) +(7-9n)*a(n-1) +2*(7n-17)*a(n-2) +10*(3-n)*a(n-3) +5*(n-4)*a(n-4) +(5-n)*a(n-5)=0. - R. J. Mathar, Nov 17 2011
Recurrence confirmed using the differential equation (6*x+2)*g+(x^5-5*x^4+10*x^3-14*x^2+9*x-1)*g'=0 satisfied by the G.f. - Robert Israel, Dec 27 2017
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (n+k) * binomial(n+2-k,3) * a(k). - Seiichi Manyama, Mar 28 2023
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