A329665
Number of meanders of length n with Motzkin-steps avoiding the consecutive steps UD, HH and DU.
Original entry on oeis.org
1, 2, 3, 6, 11, 20, 38, 72, 136, 260, 499, 958, 1847, 3572, 6917, 13422, 26097, 50808, 99049, 193354, 377857, 739148, 1447292, 2836316, 5562774, 10918180, 21444029, 42143986, 82874681, 163060540, 320996342, 632211192, 1245727488, 2455674532, 4842782497, 9554018554, 18855375593, 37224944572
Offset: 0
a(3)=6 as one has 6 meanders of length 3, namely: UUU, UUH, UHU, UHD, HUU, HUH.
A378809
Triangle read by rows: T(n,k) is the number of peak and valleyless Motzkin meanders of length n with k horizontal steps.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 5, 9, 4, 1, 1, 7, 15, 16, 5, 1, 1, 8, 27, 34, 25, 6, 1, 1, 10, 37, 76, 65, 36, 7, 1, 1, 11, 55, 124, 175, 111, 49, 8, 1, 1, 13, 69, 216, 335, 351, 175, 64, 9, 1, 1, 14, 93, 309, 675, 776, 637, 260, 81, 10, 1
Offset: 0
The triangle begins
k=0 1 2 3 4 5 6 7
n=0 1;
n=1 1, 1;
n=2 1, 2, 1;
n=3 1, 4, 3, 1;
n=4 1, 5, 9, 4, 1;
n=5 1, 7, 15, 16, 5, 1;
n=6 1, 8, 27, 34, 25, 6, 1;
n=7 1, 10, 37, 76, 65, 36, 7, 1;
...
T(3,0) = 1: UUU.
T(3,1) = 4: UUH, UHU, UHD, HUU.
T(3,2) = 3: UHH, HHU, HUH.
T(3,3) = 1: HHH.
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A088855(n,k) = {binomial(floor((n-1)/2), floor((k-1)/2))*binomial(ceil((n-1)/2),ceil((k-1)/2))}
A_xy(N) = {my(x='x+O('x^N), h = sum(n=0,N, (1/(1-y*x)^(n+1)) * (if(n<1,1,0) + sum(k=1,n, A088855(n,k)*x^(n+k-1)*(y^(k-1)) )) )); for(n=0,N-1,print(Vecrev(polcoeff(h,n))))}
A_xy(10)
A378810
Number of horizontal steps in all peak and valleyless Motzkin meanders of length n.
Original entry on oeis.org
0, 1, 4, 13, 39, 110, 300, 801, 2106, 5473, 14097, 36056, 91697, 232108, 585212, 1470557, 3684682, 9209417, 22967446, 57167993, 142051519, 352427720, 873157093, 2160579740, 5340150100, 13185150903, 32523933395, 80156852042, 197391001215, 485723767342
Offset: 0
For n = 3 we have meanders, UUU, UUH, UHU, UHD, HUU, UHH, HHU, HUH, HHH; giving a total of a(3) = 13 H steps.
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A088855(n,k) = {binomial(floor((n-1)/2), floor((k-1)/2))*binomial(ceil((n-1)/2),ceil((k-1)/2))}
A_xy(N) = {my(x='x+O('x^N), h = sum(n=0,N, (1/(1-y*x)^(n+1)) * (if(n<1,1,0) + sum(k=1,n, A088855(n,k)*x^(n+k-1)*(y^(k-1)) )) )); h}
P_xy(N) = Pol(A_xy(N), {x})
A_x(N) = {my(px = deriv(P_xy(N),y), y=1); Vecrev(eval(px))}
A_x(20)
Showing 1-3 of 3 results.
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