A191394 Number of base pyramids in all dispersed Dyck paths of length n (i.e., in all Motzkin paths of length n with no (1,0)-steps at positive heights).
0, 0, 1, 2, 6, 12, 28, 56, 121, 242, 507, 1014, 2093, 4186, 8569, 17138, 34902, 69804, 141664, 283328, 573574, 1147148, 2318010, 4636020, 9354540, 18709080, 37708672, 75417344, 151868100, 303736200, 611180252, 1222360504, 2458123705, 4916247410, 9881255187
Offset: 0
Keywords
Examples
a(4) = 6 because in HHHH, HH(UD), H(UD)H, (UD)HH, (UD)(UD), and (UUDD) we have a total of 0+1+1+1+2+1 = 6 base pyramids (shown between parentheses).
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Maple
G := 4*z^2/((1-z^2)*(1-2*z+sqrt(1-4*z^2))^2): Gser := series(G, z = 0, 38): seq(coeff(Gser, z, n), n = 0 .. 34);
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Mathematica
CoefficientList[Series[(4x^2)/((1-x^2)(1-2x+Sqrt[1-4x^2])^2), {x,0,40}], x] (* Harvey P. Dale, Jun 19 2011 *) -
Maxima
a(n):=ceiling(2*(2^n-1)/3)-sum((binomial(n-2*i+1,floor((n-2*i+1)/2))),i,1,(n+1)/2); /* Vladimir Kruchinin, Mar 15 2016 */
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PARI
x='x+O('x^50); concat([0,0], Vec(4*x^2/((1-x^2)*(1-2*x+sqrt(1-4*x^2))^2))) \\ G. C. Greubel, Mar 26 2017
Formula
a(n) = Sum_{k=0..(1 + ceiling(n/2))} k*A191392(n, k), formula clarified by G. C. Greubel.
G.f.: 4*x^2/((1-x^2)*(1-2*x+sqrt(1-4*x^2))^2).
a(n) ~ 2^(n+1)/3 * (1-sqrt(2)/sqrt(Pi*n)). - Vaclav Kotesovec, Mar 21 2014
a(n) = ceiling(2*(2^n-1)/3) - Sum_{i=1..(n+1)/2} binomial(n-2*i+1, floor((n-2*i+1)/2)) = A000975(n) - A174783(n). - Vladimir Kruchinin, Mar 15 2016
D-finite with recurrence n*a(n) -2*n*a(n-1) +(-5*n+12)*a(n-2) +2*(5*n-12)*a(n-3) +4*(n-3)*a(n-4) +8*(-n+3)*a(n-5)=0. - R. J. Mathar, Jun 14 2016
Comments