A166292 -id:A166292 - cp's OEIS frontend

A166294 Number of peaks at even level in all Dyck paths of semilength n with no UUU's and no DDD's, (U=(1,1), D=(1,-1)). These Dyck paths are counted by the secondary structure numbers (A004148).

0, 1, 4, 12, 34, 92, 242, 628, 1616, 4138, 10570, 26970, 68798, 175545, 448176, 1145058, 2927924, 7493021, 19191836, 49195806, 126205062, 324000494, 832371414, 2139802870, 5504256592, 14166936063, 36483006046, 94000206216

Offset: 1

Emeric Deutsch, Oct 12 2009

nonn

			a(3)=4 because the paths UDUDUD, UDU(UD)D, U(UD)DUD, and U(UD)(UD)D have 0 + 1 + 1 + 2 = 4 peaks at even level (shown between parentheses).

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A004148, A166291, A166292, A166293.

g := ((1-z-z^2-sqrt(1-2*z-z^2-2*z^3+z^4))*1/2)/z^3: G := z*(z-1+(1-z+z^2)*g)/((1-z-z^2)*(1-z-z^2-2*z^3*g)): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 1 .. 30);

Rest[CoefficientList[Series[x*(x-1+(1-x+x^2)*((1-x-x^2-Sqrt[1-2*x-x^2-2*x^3+x^4])*1/2)/x^3)/((1-x-x^2)*(1-x-x^2-2*x^3*((1-x-x^2-Sqrt[1-2*x-x^2-2*x^3+x^4])*1/2)/x^3)), {x, 0, 20}], x]] (* Vaclav Kotesovec, Mar 20 2014 *)

a(n) = Sum_{k=0..n-1} k*A166293(n,k).
G.f.: G=z*(z - 1 + (1 - z + z^2)*g(z))/((1 - z - z^2)*(1 - z - z^2 - 2*z^3*g(z))), where g=g(z) satisfies g = 1 + z*g + z^2*g + z^3*g^2.
a(n) ~ sqrt(55 + 123/sqrt(5)) * (3+sqrt(5))^n / (sqrt(Pi*n) * 2^(n+5/2)). - Vaclav Kotesovec, Mar 20 2014
Equivalently, a(n) ~ phi^(2*n + 5) / (4 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021

A166291 Triangle read by rows: T(n,k) is the number of Dyck paths with no UUU's and no DDD's, of semilength n and having k peaks at odd level (0<=k<=n; U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 2, 3, 0, 1, 3, 6, 3, 4, 0, 1, 5, 10, 12, 4, 5, 0, 1, 8, 20, 22, 20, 5, 6, 0, 1, 13, 36, 52, 40, 30, 6, 7, 0, 1, 21, 66, 104, 109, 65, 42, 7, 8, 0, 1, 34, 118, 214, 240, 200, 98, 56, 8, 9, 0, 1, 55, 210, 421, 549, 481, 335, 140, 72, 9, 10, 0, 1, 89, 370

Offset: 0

Emeric Deutsch, Oct 12 2009

Sum of entries in row n = A004148(n+1) (the secondary structure numbers).
T(n,0)=A000045(n-1) (the Fibonacci numbers).
Sum(k*T(n,k), k=0..n)=A166292(n).

			T(4,2)=3 because we have (UD)(UD)UUDD, (UD)UUDD(UD), and UUDD(UD)(UD) (the odd level peaks are shown between parentheses).
Triangle starts:
1;
0,1;
1,0,1;
1,2,0,1;
2,2,3,0,1;
3,6,3,4,0,1.

Cf. A004148, A000045, A166292, A166293, A166294

p1 := -G+1+t*z*G+s*z^2*G+s^2*z^3*H*G: p2 := subs({t = s, s = t, G = H, H = G}, p1): r := resultant(p1, p2, H): G := RootOf(subs(s = 1, r), G): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, j), j = 0 .. n) end do; yields sequence in triangular form

The trivariate g.f. G=G(t,s,z), where z marks semilength, t marks odd-level peaks and s marks even-level peaks, satisfies G = 1 + tzG + sz^2*G + s^2*z^3*HG, where H=G(s,t,z) (interchanging t and s and eliminating H, one obtains G(t,s,z); see the Maple program).

A166293 Triangle read by rows: T(n,k) is the number of Dyck paths with no UUU's and no DDD's, of semilength n and having k peaks at even level (0<=k<=n-1; U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 12, 5, 1, 1, 6, 22, 28, 18, 6, 1, 1, 7, 35, 59, 50, 25, 7, 1, 1, 8, 54, 114, 124, 80, 33, 8, 1, 1, 9, 82, 210, 279, 226, 119, 42, 9, 1, 1, 10, 124, 374, 592, 576, 375, 168, 52, 10, 1, 1, 11, 188, 653, 1199, 1374, 1062, 582, 228, 63

Offset: 1

Emeric Deutsch, Oct 12 2009

Sum of entries in row n = A004148(n+1) (the secondary structure numbers).
T(n,0)=1.
Sum(k*T(n,k), k=0..n-1)=A166294(n).

			T(4,2)=3 because we have UDU(UD)(UD)D, U(UD)(UD)DUD, and U(UD)DU(UD)D (the even-level peaks are shown between parentheses).
Triangle starts:
1;
1,1;
1,2,1;
1,3,3,1;
1,4,7,4,1;
1,5,13,12,5,1.

Cf. A004148, A166291, A166292, A166294

p1 := -G+1+t*z*G+s*z^2*G+s^2*z^3*H*G: p2 := subs({t = s, s = t, G = H, H = G}, p1): r := resultant(p1, p2, H): G := RootOf(subs(t = 1, r), G): Gser := simplify(series(G, z = 0, 15)): for n to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 12 do seq(coeff(P[n], s, j), j = 0 .. n-1) end do; # yields sequence in triangular form

The trivariate g.f. G=G(t,s,z), where z marks semilength, t marks odd-level peaks and s marks even-level peaks, satisfies G = 1 + tzG + sz^2*G + s^2*z^3*HG, where H=G(s,t,z) (interchanging t and s and eliminating H, one obtains G(t,s,z); see the Maple program).

cp's OEIS Frontend

A166294 Number of peaks at even level in all Dyck paths of semilength n with no UUU's and no DDD's, (U=(1,1), D=(1,-1)). These Dyck paths are counted by the secondary structure numbers (A004148).

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A166291 Triangle read by rows: T(n,k) is the number of Dyck paths with no UUU's and no DDD's, of semilength n and having k peaks at odd level (0<=k<=n; U=(1,1), D=(1,-1)).

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A166293 Triangle read by rows: T(n,k) is the number of Dyck paths with no UUU's and no DDD's, of semilength n and having k peaks at even level (0<=k<=n-1; U=(1,1), D=(1,-1)).

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Formula