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A071719
Expansion of (1+x^2*C)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
2
1, 4, 15, 53, 185, 647, 2277, 8073, 28834, 103700, 375326, 1366290, 4999717, 18382405, 67877025, 251615745, 936047790, 3493585920, 13077995730, 49091198550, 184742369370, 696863109774, 2634345976818, 9978753842378
OFFSET
0,2
LINKS
FORMULA
Conjecture: -3*(n+4)*(3*n-1)*a(n) +5*(n+2)*(7*n+3)*a(n-1) +2*(2*n-3)*(n-7)*a(n-2)=0. - R. J. Mathar, Aug 25 2013
Conjecture confirmed using the differential equation (4*x^4+35*x^3-9*x^2)*g'' + (-14*x^3+190*x^2-42*x)*g' + (-10*x^2+150*x+12)*g = 30*x+12 satisfied by the G.f. - Robert Israel, Jun 07 2018
a(n) ~ 37 * 4^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 23 2019
MAPLE
f:= gfun:-rectoproc({-3*(n+4)*(3*n-1)*a(n) +5*(n+2)*(7*n+3)*a(n-1) +2*(2*n-3)*(n-7)*a(n-2)=0, a(0)=1, a(1)=4}, a(n), remember):
map(f, [$0..40]); # Robert Israel, Jun 07 2018
MATHEMATICA
(1 + c x^2) c^4 /. c -> (1 - (1 - 4x)^(1/2))/(2x) + O[x]^24 // CoefficientList[#, x]& (* Jean-François Alcover, Oct 23 2019 *)
CROSSREFS
Sequence in context: A367818 A303271 A289802 * A370034 A289927 A164619
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 06 2002
STATUS
approved