A118926
Invariant column vector V under matrix product A104546*V = V: a(n) = Sum_{k=0,[n/2]} A104546(n,k)*a(k).
Original entry on oeis.org
1, 2, 7, 28, 125, 598, 3007, 15708, 84585, 466954, 2632167, 15103676, 88012801, 519848442, 3107443803, 18774545752, 114527169657, 704731976138, 4370943547471, 27306560735812, 171728169545661, 1086605771091766
Offset: 0
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{a(n)=local(G=1+x+x*O(x^n));if(n==0,1, for(i=0,n,G=1+x*G+x*G*(G+(y-1)*x/(1-x))); sum(k=0,n\2,a(k)*polcoeff(polcoeff(G+y*O(y^k),n,x),k,y)))}
A104547
Number of Schroeder paths of length 2n having no UHD, UHHD, UHHHD, ..., where U=(1,1), D=(1,-1), H=(2,0).
Original entry on oeis.org
1, 2, 5, 16, 60, 245, 1051, 4660, 21174, 98072, 461330, 2197997, 10585173, 51443379, 251982793, 1242734592, 6165798680, 30754144182, 154123971932, 775669589436, 3918703613376, 19866054609754, 101029857327802, 515275408644773
Offset: 0
a(2)=5 because we have HH, HUD, UDH, UDUD and UUDD (UHD does not qualify).
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R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-2*x+2*x^2 - Sqrt(1-8*x+16*x^2-12*x^3+4*x^4))/(2*x*(1-x)) )); // G. C. Greubel, Jan 02 2023
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CoefficientList[Series[(1-2*x+2*x^2 -Sqrt[1-8*x+16*x^2-12*x^3+4*x^4] )/(2*x*(1-x)), {x,0,40}], x] (* G. C. Greubel, Jan 02 2023 *)
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{a(n)=polcoeff(2*(1-x)/(1-2*x+2*x^2 + sqrt(1-8*x+16*x^2-12*x^3+4*x^4+x*O(x^n))),n)} \\ Paul D. Hanna, May 17 2006
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def A104547_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P( (1-2*x+2*x^2 - sqrt(1-8*x+16*x^2-12*x^3+4*x^4))/(2*x*(1-x)) ).list()
A104547_list(40) # G. C. Greubel, Jan 02 2023
A118927
Self-convolution square-root of A118926.
Original entry on oeis.org
1, 1, 3, 11, 47, 219, 1083, 5597, 29933, 164547, 925289, 5303409, 30898069, 182587159, 1092498017, 6609496707, 40383429717, 248940496629, 1546962207417, 9683696676693, 61024927547241, 386936560902269, 2467341175475253
Offset: 0
A118926 satisfies: A118926(n) = Sum_{k=0,[n/2]} A104546(n,k)*A118926(k),
where A104546(n,k) is the number of Schroeder paths of length 2n and having k platforms. Is this sequence always integer?
Showing 1-3 of 3 results.
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