cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A166292 Number of peaks at odd 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).

Original entry on oeis.org

0, 1, 2, 5, 12, 29, 72, 181, 460, 1178, 3030, 7815, 20188, 52193, 134992, 349205, 903398, 2337135, 6046310, 15642402, 40469824, 104708914, 270937964, 701129755, 1814581514, 4696886211, 12159165336, 31481922733, 81523933604, 211143257951
Offset: 0

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Author

Emeric Deutsch, Oct 12 2009

Keywords

Examples

			a(3)=5 because the paths (UD)(UD)(UD), (UD)UUDD, UUDD(UD), and UUDUDD have 3 + 1 + 1 + 0 peaks at odd level (shown between parentheses).

Links

Crossrefs

Cf. A004148, A166291, A166293, A166294

Programs

  • Maple
    g := ((1-z-z^2-sqrt(1-2*z-z^2-2*z^3+z^4))*1/2)/z^3: G := (z^2+3*z-2+(z^4+z^3-z^2-4*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 = 0 .. 30);
  • Mathematica
    CoefficientList[Series[(x^2+3*x-2+(x^4+x^3-x^2-4*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 *)

Formula

a(n) = Sum_{k=0..n} k*A166291(n,k).
G.f.: G=(z^2 + 3*z - 2 + (z^4 + z^3 - z^2 - 4*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
D-finite with recurrence -(n+3)*(12263959*n-453745718)*a(n) +(-12263959*n^2-1898630196*n-3269930920)*a(n-1) +2*(201299201*n^2-33603137*n-693679841)*a(n-2) +4*(-226348358*n^2+2057096353*n-415271997)*a(n-3) +(282398701*n^2-4452810911*n+4125507350)*a(n-4) +(46943591*n^2-3487699726*n+8812781352)*a(n-5) +4*(214523727*n^2-2147935054*n+5837738425)*a(n-6) +2*(-101430217*n^2+3147700949*n-14145800266)*a(n-7) -5*(76453537*n-336152052)*(n-7)*a(n-8) +(n-8)*(126836791*n-1033250150)*a(n-9)=0. - R. J. Mathar, Jul 26 2022

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

Views

Author

Emeric Deutsch, Oct 12 2009

Keywords

Comments

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).

Examples

			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.

Crossrefs

Cf. A004148, A000045, A166292, A166293, A166294

Programs

  • Maple
    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

Formula

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

Views

Author

Emeric Deutsch, Oct 12 2009

Keywords

Comments

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).

Examples

			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.

Crossrefs

Cf. A004148, A166291, A166292, A166294

Programs

  • Maple
    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

Formula

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).
Showing 1-3 of 3 results.

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