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A257390
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Number of 4-Motzkin paths of length n with no level steps at even level.
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1
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1, 0, 1, 4, 18, 80, 357, 1596, 7150, 32096, 144362, 650568, 2937316, 13286368, 60205805, 273290988, 1242639446, 5659468736, 25816338046, 117945079736, 539646216188, 2472638868960, 11345220210658, 52124831171544, 239792244636876, 1104495824173376
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OFFSET
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0,4
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LINKS
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FORMULA
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a(n) = Sum_{i=0..floor(n/2)}4^(n-2i)*C(i)*binomial(n-i-1,n), where C(i) is the i-th Catalan number A000108.
G.f.: (1-4*x-sqrt((1-4*x)*(1-4*x-4*x^2)))/(2*x^2).
a(n) ~ 2^(n+3/4) * (1+sqrt(2))^(n+1/2) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 22 2015
a(n) = ((-16*n + 40)*a(n-3) + (-12*n+12)*a(n-2) +(8*n+4)*a(n-1))/(n+2). - Robert Israel, Apr 22 2015
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MAPLE
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rec:= a(n) = ((-16*n + 40)*a(n-3) + (-12*n+12)*a(n-2) +(8*n+4)*a(n-1))/(n+2):
f:= gfun:-rectoproc({rec, a(0)=1, a(1)=0, a(2)=1}, a(n), remember):
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MATHEMATICA
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CoefficientList[Series[(1-4*x-Sqrt[(1-4*x)*(1-4*x-4*x^2)])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 22 2015 *)
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PROG
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(PARI) x='x+O('x^50); Vec((1-4*x-sqrt((1-4*x)*(1-4*x-4*x^2)))/(2*x^2)) \\ G. C. Greubel, Apr 08 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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