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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A190170 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n and having k UHD's starting at level 0; here U=(1,1), H=(1,0), and D=(1,-1).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 5, 3, 12, 4, 1, 27, 7, 3, 60, 16, 6, 135, 39, 10, 1, 309, 92, 18, 4, 717, 212, 39, 10, 1680, 488, 94, 20, 1, 3966, 1135, 228, 39, 5, 9423, 2670, 543, 84, 15, 22518, 6336, 1282, 200, 35, 1, 54091, 15132, 3036, 492, 75, 6, 130540, 36327, 7245, 1203, 166, 21
Offset: 0

Views

Author

Emeric Deutsch, May 06 2011

Keywords

Comments

Row n contains 1+floor(n/3) entries.
Sum of entries in row n = A004148(n).
T(n,0)=A190171(n).
Sum(k*T(n,k),k>=0)=A089735(n-3).

Examples

			T(6,2)=1 because we have UHDUHD.
Triangle starts:
1;
1;
1;
1,1;
2,2;
5,3;
12,4,1;
27,7,3;

Crossrefs

Cf. A004148, A190171, A089735.

Programs

  • Maple
    p1 := G-1-z*G-z^2*G*(S-1-z+t*z): p2 := S-1-z*S-z^2*S*(S-1): r := resultant(p1, p2, S): g := RootOf(r, G): Gser := simplify(series(g, z = 0, 21)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 17 do seq(coeff(P[n], t, j), j = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form

Formula

G.f. G=G(t,z) is obtained by elimitaing S from the equations G=1+zG+z^2*G(S-1-z+tz) and S=1+zS+z^2*S(S-1).

A191579 Triangular array related to continued fractions of square root of (N^2 - 1) for N>1, apparently containing A004148 and summing to A091964.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 4, 6, 6, 4, 1, 8, 13, 13, 10, 5, 1, 17, 28, 30, 24, 15, 6, 1, 37, 62, 69, 59, 40, 21, 7, 1, 82, 140, 160, 144, 105, 62, 28, 8, 1, 185, 320, 375, 350, 271, 174, 91, 36, 9, 1, 423, 740, 885, 852, 690, 474, 273, 128, 45, 10, 1
Offset: 1

Views

Author

Kenneth J Ramsey, Jun 07 2011

Keywords

Comments

The row sums of this triangle seems to be A091964 (verified to 12 terms), cf. diagonal sums of the triangle A124428. The 1st column seems to be A004148. The 2nd and 3rd column seems to be A089735, and A098075 (verified to 10 terms).
As each of these sequence is related to enumeration of RNA molecule structures, but was generated independently by reference to square array A192062 (re continued fractions for square roots of n^2-1 for n>1, see comments in the example below), it could be interesting to check this further for a relationship. As Mathar noted, this triangle appears identical to A097724. - edited by Kenneth J Ramsey, Oct 25 2012
Is this (apart from offsets) the same as A097724? - R. J. Mathar, Aug 01 2011

Examples

			The triangle begins
1;
1, 1;
1, 2, 1;
2, 3, 3, 1;
4, 6, 6, 4, 1;
8, 13, 13, 10, 5, 1;
17, 28, 30, 24, 15, 6, 1;
37, 62, 69, 59, 40, 21, 7, 1;
82, 140, 160, 144, 105, 62, 28, 8, 1;
185, 320, 375, 350, 271, 174, 91, 36, 9, 1;
423, 740, 885, 852, 690, 474, 273, 128, 45, 10, 1;
...
The 4th row is 2,3,3,1 because the 2nd,4th,6th and 8th terms of columns j = 1-5 of square array T(i,j) A192062  form the 4*5 matrix {{1,3,8,21},{1,4,15,56},{1,5,24,115},{1,6,35,204},{1,7,48,329}}. Solving the resulting system of linear equations results in the identities:
2*1 + 3*3 + 3*8 + 1*21 = 56 = T(8,2) of A192062
2*1 + 3*4 + 3*15+ 1*56 = 115 = T(8,3) of A192062
2*1 + 3*5 + 3*24 + 1*115 = 204 = T(8,4) of A192062
2*1 + 3*6 + 3*35 + 1*204 = 329 = T(8,5) of A192062

Crossrefs

Cf. A091964, A124428, A004148, A089735, A098075, A192062.

Formula

The only way I know to generate this triangle is by reference to the square array A192062. The columns of that array, T(i,j) are such that for any given i>0, each term T(i,2*n) equals the sum as k = 1 to n, T(i-1,2*k)*C_k where C_k is the k th term of the n th row of this triangle. So solving the system of linear equations for each n > 0 gives the n th row of this triangle.

A187260 Number of uh^jd's for some j>0, starting at level 0, where u=(1,1), h=(1,0), and d=(1,-1), in all peakless Motzkin paths of length n (can be easily expressed using RNA secondary structure terminology).

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 12, 25, 53, 115, 255, 575, 1315, 3043, 7111, 16756, 39766, 94961, 228003, 550081, 1332839, 3241930, 7913028, 19375635, 47579847, 117149125, 289142441, 715253644, 1773011502, 4403539181, 10956537307, 27307002454, 68164324150, 170404155586, 426584025250, 1069289177950
Offset: 0

Views

Author

Emeric Deutsch, May 05 2011

Keywords

Comments

The terms a(n), starting from n=3, are the partial sums of the sequence A089735.

Examples

			a(4)=3 because the 4 (=A004148(4)) peakless Motzkin paths of length 4, namely hhhh, h(uhd), (uhd)h, and (uhhd) contain 0+1+1+1 subwords of type uh^jd for some j>0, starting at level 0 (shown between parentheses).

Crossrefs

Cf. A098071, A004148, A089735.

Programs

  • Maple
    eq := g = 1+z*g+z^2*g*(g-1): g := RootOf(eq, g): F := z^3*g^2/(1-z): Fser := series(F, z = 0, 38): seq(coeff(Fser, z, n), n = 0 .. 35);
  • Mathematica
    CoefficientList[Series[(-1 + x - x^2 + Sqrt[(1 + (-3 + x)*x)*(1 + x + x^2)])^2 / (4*(1 - x)*x), {x, 0, 40}], x] (* Vaclav Kotesovec, May 29 2022 *)

Formula

G.f.: z^3*g^2/(1-z), where g=1+z*g+z^2*g*(g-1).
a(n) = Sum_{k>=0} k*A098071(n,k).
From Vaclav Kotesovec, May 29 2022: (Start)
G.f.: (-1 + x - x^2 + sqrt((1 + (-3 + x)*x) * (1 + x + x^2)))^2 / (4*(1-x)*x).
a(n) ~ 5^(1/4) * phi^(2*n-1) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. (End)
D-finite with recurrence (n+1)*a(n) +(-4*n+1)*a(n-1) +(5*n-8)*a(n-2) +(-5*n+18)*a(n-3) +(5*n-22)*a(n-4) +(-5*n+32)*a(n-5) +(4*n-31)*a(n-6) +(-n+9)*a(n-7)=0. - R. J. Mathar, Jul 22 2022
Showing 1-3 of 3 results.

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