A108431
Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have k hills (a hill is either a ud or a Udd starting at the x-axis).
Original entry on oeis.org
1, 0, 2, 6, 0, 4, 34, 24, 0, 8, 274, 136, 72, 0, 16, 2266, 1168, 408, 192, 0, 32, 19738, 9880, 3720, 1088, 480, 0, 64, 177642, 87840, 32088, 10496, 2720, 1152, 0, 128, 1640050, 802216, 291048, 92096, 27680, 6528, 2688, 0, 256, 15445690, 7492240
Offset: 0
Example T(2,2)=4 because we have udud, udUdd, Uddud and UddUdd.
Triangle begins:
1;
0,2;
6,0,4;
34,24,0,8;
...
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A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: G:=1/(1-z*A+2*z-2*z*t-z*A^2): Gserz:=simplify(series(G,z=0,12)): P[0]:=1: for n from 1 to 10 do P[n]:=sort(coeff(Gserz,z^n)) od: for n from 0 to 9 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; expand(`if`(y<0 or y>x, 0,
`if`(x=0, 1, b(x-1, y-1, t)*`if`(t and y=1, z, 1)+
b(x-1, y+2, is(y=0))+b(x-2, y+1, is(y=0)))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..n))(b(3*n, 0, false)):
seq(T(n), n=0..10); # Alois P. Heinz, Oct 06 2015
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b[x_, y_, t_] := b[x, y, t] = Expand[If[y<0 || y>x, 0, If[x == 0, 1, b[x-1, y-1, t]*If[t && y == 1, z, 1] + b[x-1, y+2, y == 0] + b[x-2, y+1, y == 0]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, n}]][b[ 3*n, 0, False]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 25 2015, after Alois P. Heinz *)
A108432
Number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have no hills (a hill is either a ud or a Udd starting at the x-axis).
Original entry on oeis.org
1, 0, 6, 34, 274, 2266, 19738, 177642, 1640050, 15445690, 147813706, 1433309194, 14052298690, 139063589370, 1387288675002, 13936344557354, 140859338668306, 1431424362057018, 14616361066692778, 149892742974500042, 1543146417012350050, 15942622531081651578
Offset: 0
a(2)=6 because we have uudd, uUddd, Ududd, UdUddd, Uuddd and UUdddd.
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g:=1/(1+2*z-z*A-z*A^2): A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3:gser:=series(g,z=0,27): 1,seq(coeff(gser,z^n),n=1..24);
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
`if`(x=0, 1, `if`(t and y=1, 0, b(x-1, y-1, t))+
b(x-1, y+2, is(y=0))+b(x-2, y+1, is(y=0))))
end:
a:= n-> b(3*n, 0, false):
seq(a(n), n=0..25); # Alois P. Heinz, Oct 06 2015
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CoefficientList[Series[9/(3 + 18*x + 2*(3+x)*Cos[2/3*ArcSin[Sqrt[x]*(18+x)/(3+x)^(3/2)]] - 2*x*Sqrt[(3+x)/x]*Sin[1/3*ArcSin[Sqrt[x]*(18+x)/(3+x)^(3/2)]]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 17 2014 *)
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{a(n)=local(y=1); for(i=1, n, y=-(-1 + 6*x*y - 5*x*y^2 - 12*x^2*y^2 - x*y^3 + 6*x^2*y^3 + 8*x^3*y^3) + (O(x^n))^3); polcoeff(y, n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 17 2014
A108434
Number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have no hills of the form ud (a hill is either a ud or a Udd starting at the x-axis).
Original entry on oeis.org
1, 1, 7, 47, 361, 2977, 25775, 231103, 2127409, 19990241, 190957559, 1848911279, 18104425561, 178975914433, 1783843502047, 17906040994559, 180858717257185, 1836792828317761, 18745545101801063, 192145823547338927
Offset: 0
a(2)=7 because we have uudd, uUddd, UddUdd, Ududd, UdUddd, Uuddd and UUdddd.
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g:=1/(1+z-z*A-z*A^2): A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3:gser:=series(g,z=0,27): 1,seq(coeff(gser,z^n),n=1..24);
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{a(n)=local(y=1+x); for(i=1, n, y=-(-1 + 3*x*y - 3*x*(1+x)*y^2 + x*(-1+2*x+x^2)*y^3) + (O(x^n))^3); polcoeff(y, n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 17 2014
Showing 1-3 of 3 results.
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