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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A190167 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n having a total of k (1,0)-steps at levels 1,3,5,... .

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 3, 2, 1, 4, 6, 4, 2, 1, 7, 12, 10, 5, 2, 1, 12, 24, 23, 14, 6, 2, 1, 21, 48, 52, 36, 18, 7, 2, 1, 38, 96, 115, 90, 51, 22, 8, 2, 1, 70, 193, 254, 217, 138, 68, 26, 9, 2, 1, 130, 388, 559, 522, 358, 196, 87, 30, 10, 2, 1, 243, 782, 1220, 1240, 926, 542, 264, 108, 34, 11, 2, 1
Offset: 0

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Author

Emeric Deutsch, May 06 2011

Keywords

Comments

Row n has n-1 entries (n>=3).
Sum of entries in row n is A004148(n) (the RNA secondary structure numbers).
T(n,0)=A190168(n).
Sum(kT(n,k),k>=0)=A190169(n).
The trivariate g.f. H(t,s,z), where t (s) marks (1,0)-steps at even (odd) levels and z marks length, satisfies
z^2(1-tz+z^2)H^2-(1-tz+z^2)(1-sz+z^2)H+1-sz+z^2=0.

Examples

			T(5,2)=2 because we have huh'h'd and uh'h'dh, where u=(1,1), h=(1,0), d=(1,-1) (the odd-level h-steps are marked).
Triangle starts:
1;
1;
1;
1,1;
1,2,1;
2,3,2,1;
4,6,4,2,1;
7,12,10,5,2,1;

Crossrefs

Cf. A004148, A190164, A190168, A190169

Programs

  • Maple
    eq:=z^2*(1-z+z^2)*G^2-(1-z+z^2)*(1-s*z+z^2)*G+1-s*z+z^2 = 0: g:= RootOf(eq, G): Gser:= simplify(series(g, z = 0, 17)): for n from 0 to 13 do P[n] := sort(expand(coeff(Gser, z, n))) end do: 1; 1; for n from 0 to 13 do seq(coeff(P[n], s, k), k = 0 .. n-2) end do; # yields sequence in triangular form

Formula

G.f. = G = G(s,z) satisfies the equation z^2*(1-z+z^2)G^2-(1-z+z^2)(1-sz+z^2)G+1-sz+z^2=0.

A190165 Number of peakless Motzkin paths of length n having no (1,0)-steps at levels 0,2,4,... .

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 22, 41, 76, 142, 268, 509, 971, 1861, 3583, 6925, 13430, 26128, 50980, 99735, 195594, 384454, 757256, 1494465, 2954715, 5851677, 11607348, 23058492, 45870685, 91371464, 182231978, 363871075, 727364502, 1455503056, 2915461721, 5845386764, 11730347948
Offset: 0

Views

Author

Emeric Deutsch, May 06 2011

Keywords

Comments

a(n) = A190164(n,0).

Examples

			a(6)=2 because we have uhduhd and uhhhhd, where u=(1,1), h=(1,0), d=(1,-1).

Crossrefs

Cf. A190164, A190167, A190168

Programs

  • Maple
    eq := z^2*(1+z^2)*G^2-(1+z^2)*(1-z+z^2)*G+1-z+z^2 =0: g:=RootOf(eq,G): Gser:=series(g,z=0,46): seq(coeff(Gser,z,n),n=0..40);

Formula

G.f. G=G(z) satisfies the equation z^2*(1+z^2)G^2 - (1+z^2)(1-z+z^2)G + 1-z+z^2=0.
D-finite with recurrence +(n+2)*a(n) +(-2*n-1)*a(n-1) +(n+1)*a(n-2) +(-2*n+1)*a(n-3) +(-2*n+11)*a(n-5) +(n-7)*a(n-6) +(-2*n+13)*a(n-7) +(n-8)*a(n-8)=0. - R. J. Mathar, Jul 24 2022

A190166 Number of (1,0)-steps at levels 0,2,4,... in all peakless Motzkin paths of length n.

Original entry on oeis.org

0, 1, 2, 3, 6, 14, 34, 83, 202, 495, 1224, 3046, 7616, 19115, 48130, 121527, 307602, 780244, 1982834, 5047377, 12867438, 32847357, 83952780, 214806750, 550170300, 1410412561, 3618785462, 9292203549, 23877482490, 61397367692, 157972743178, 406693829059, 1047585820586, 2699811117189
Offset: 0

Views

Author

Emeric Deutsch, May 06 2011

Keywords

Comments

a(n)=Sum(k*A190164(n,k),k>=0).
a(n)=A110236(n) - A190169(n).

Examples

			a(4)=6 because in h'h'h'h', h'uhd, uhdh', and uhhd, where u=(1,1), h=(1,0), d=(1,-1), we have 4+1+1+0 h-steps at even levels (marked).

Crossrefs

Cf. A190164, A110236, A190169, A004148.

Programs

  • Maple
    G := z/((1-z+z^2)*sqrt((1+z+z^2)*(1-3*z+z^2))): Gser := series(G,z=0,36): seq(coeff(Gser,z,n),n=0..33);

Formula

G.f. = z/[(1-z+z^2)sqrt((1+z+z^2)(1-3z+z^2))].
Conjecture: (-n+1)*a(n) +(3*n-4)*a(n-1) +2*(-n+1)*a(n-2) +3*(n-2)*a(n-3) +2*(-n+3)*a(n-4) +(3*n-8)*a(n-5) +(-n+3)*a(n-6)=0. - R. J. Mathar, Apr 09 2019
a(n) ~ phi^(2*n+2) / (4 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 29 2022
Showing 1-3 of 3 results.

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