A004148 -id:A004148 - cp's OEIS frontend

A097860 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n with k peaks (n>=0, 0<=k<=floor(n/2)).

1, 1, 1, 1, 2, 2, 4, 4, 1, 8, 10, 3, 17, 24, 9, 1, 37, 58, 28, 4, 82, 143, 81, 16, 1, 185, 354, 231, 60, 5, 423, 881, 653, 205, 25, 1, 978, 2204, 1824, 676, 110, 6, 2283, 5534, 5058, 2164, 435, 36, 1, 5373, 13940, 13946, 6756, 1631, 182, 7, 12735, 35213, 38262, 20710

Offset: 0

Emeric Deutsch, Sep 01 2004

Row sums are the Motzkin numbers (A001006). Column 0 gives A004148.

This triangle is the Motzkin path equivalent to the Narayana numbers (A001263). - Dan Drake, Feb 17 2011

			Triangle starts:
   1;
   1;
   1,  1;
   2,  2;
   4,  4, 1;
   8, 10, 3;
  17, 24, 9, 1;
  ...
Row n has 1+floor(n/2) terms.
T(4,1)=4 because (UD)HH, H(UD)H, HH(UD) and U(UD)D are the only Motzkin paths of length 4 with 1 peak (here U=(1,1), H=(1,0) and D=(1,-1)); peaks are shown between parentheses.

Alois P. Heinz, Rows n = 0..200, flattened
Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, Motzkin and Catalan Tunnel Polynomials, J. Int. Seq., Vol. 21 (2018), Article 18.8.8.
Dan Drake and Ryan Gantner, Generating functions for plateaus in Motzkin paths, J. of the Chungcheong Math. Soc., Vol 25, No. 3, p. 475, August 2012.
Zhuang, Yan. A generalized Goulden-Jackson cluster method and lattice path enumeration, Discrete Mathematics 341.2 (2018): 358-379. Also arXiv: 1508.02793v2.

Cf. A001006, A004148, A001263, A097885.

eq:=G=1+z*G+z^2*G*(G-1+t):sol:=solve(eq,G): G:=sol[2]: Gser:=simplify(series(G,z=0,16)): P[0]:=1: for n from 1 to 13 do P[n]:=sort(coeff(Gser,z^n)) od: seq(seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)),n=0..13);
# Alternatively
A097860_row := proc(n) local w,f,p,i;
w := 1-x+x^2-x^2*t; f := (w-sqrt(w^2-4*x^2))/(2*x^2);
p := simplify(coeff(series(f,x,3+2*n),x,n));
seq(coeff(p,t,i), i=0..iquo(n,2)) end:
seq(print(A097860_row(n)), n=0..7); # Peter Luschny, Nov 14 2014
# third Maple program:
b:= proc(x, y, t) option remember; expand(`if`(y<0 or y>x, 0,
     `if`(x=0, 1, b(x-1, y, 1)+b(x-1, y-1, 1)*t+b(x-1, y+1, z))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0, 1)):
seq(T(n), n=0..15);  # Alois P. Heinz, Feb 01 2019

gf = With[{w = 1 - x + x^2 - x^2*t}, (w - Sqrt[w^2 - 4*x^2])/(2*x^2)];
cx[n_] := cx[n] = SeriesCoefficient[gf, {x, 0, n}];
T[n_, k_] := SeriesCoefficient[cx[n], {t, 0, k}];
Table[T[n, k], {n, 0, 14}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Dec 04 2017, after Peter Luschny *)

G.f. G = G(t, z) satisfies G = 1+z*G+z^2*G*(G-1+t).

G.f. has explicit form G(x,t) = (w-sqrt(w^2-4*x^2))/(2*x^2) with w = 1-x+x^2-x^2*t. (Drake and Ganter, Th. 6) - Peter Luschny, Nov 14 2014

A098056 Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n containing k subwords of the type uh^ju, dH^jd, or dh^ju for some j>0, where u=(1,1), d=(1,-1) and h=(1,0) (can be easily expressed using RNA secondary structure terminology).

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 15, 2, 27, 9, 1, 48, 29, 5, 84, 80, 21, 147, 198, 74, 4, 257, 463, 230, 27, 1, 451, 1033, 667, 125, 7, 796, 2235, 1811, 488, 43, 1413, 4727, 4694, 1676, 219, 6, 2526, 9828, 11700, 5317, 946, 54, 1, 4544, 20192, 28252, 15813, 3696, 326, 9, 8226, 41100

Offset: 0

Emeric Deutsch, Sep 11 2004

Row sums are the RNA secondary structure numbers (A004148).
T(n,0) = A098057(n).
Sum(k*T(n,k),k>=0) = A187259(n).

			Triangle starts:
  1;
  1;
  1;
  2;
  4;
  8;
  15,2;
  27,9,1;
  48,29,5;
  84.80,21;
  147,198,74,7;
  ...
It seems that the number r(n) of terms in row n>=3 is given by r(n)=n/2-1 if n=2 (mod 4) and r(n)=2*round(n/4)-1 otherwise (here round(m) is the nearest integer to m).
T(7,1)=9 because we have h(uhu)hdd, (uhhu)hdd, (uhu)hhdd, (uhu)hddh, uh(dhu)hd and the reflections of the first four paths in a vertical axis; here u=(1,1), h=(1,0), d=(1,-1) and the pertinent subwords are shown between parentheses.

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.
M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08; Sem. Loth. Comb. B08l (1984) 79-86.

Cf. A041048, A098057, A187259.

G.f.=G=G(t, z) satisfies G = 1 + zG + z^2*[H + 2tzH/(1-z)+t^2*z^2*H/(1-z)^2+ z/(1-z)][G-(1-t)zH/(1-z)^2], where H=(1-z)^2*G-1+z.
The 4-variate g.f. G(t,s,v,z) of peakless Motzkin paths, where t, s, v mark subwords of the types uH^ju, dH^jd, dH^ju, respectively, and z marks length, satisfies the equation
G = 1+zG+z^2*[H + (t+s)zH/(1-z)+tsz^2*H/(1-z)^2+z/(1-z)][G-(1-v)zH/(1-z)^2],
where H = (1-z)[(1-z)G-1]. As special cases we get the current sequence A098056 and the sequences A097777 and A098083.

A098057 Number of peakless Motzkin paths with no U H^j U, no D H^j D and no D H^jU (j>0), where U=(1,1), D=(1,-1) and H=(1,0) (can be easily expressed using RNA secondary structure terminology).

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 15, 27, 48, 84, 147, 257, 451, 796, 1413, 2526, 4544, 8226, 14978, 27417, 50434, 93183, 172865, 321857, 601263, 1126644, 2116968, 3987960, 7530200, 14249649, 27019301, 51327965, 97676156, 186177568, 355406479, 679425009

Offset: 0

Emeric Deutsch, Sep 11 2004

nonn

			a(4)=4 because we have HHHH, UHDU, HUHD and UHHD; a(6)=15 because from all 17 peakless Motzkin paths of length 6 (see A004148) only (UHU)HDD and UUH(DHD) do not qualify.

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.
M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire, Sem. Loth. Comb. B08l (1984) 79-86.

Cf. A004148.

G.f.: [1-z+z^2-4z^3+2z^4-sqrt(1-2z-z^2+2z^3+z^4-4z^5+4z^6)]/[2z^2*(1-z)^3].

D-finite with recurrence (n+2)*a(n) +(-5*n-7)*a(n-1) +2*(4*n+5)*a(n-2) +(-2*n-13)*a(n-3) +3*(-2*n+5)*a(n-4) +18*(1)*a(n-5) +(17*n-101)*a(n-6) +(-25*n+154)*a(n-7) +2*(8*n-53)*a(n-8) +4*(-n+7)*a(n-9)=0. - R. J. Mathar, Jul 26 2022

A098071 Triangle read by rows: T(n,k) is number of peakless Motzkin paths of length n and having k uhh...hd's starting at level 0, where u=(1,1), h=(1,0) and d=(1,-1) (can be easily expressed using RNA secondary structure terminology).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 2, 6, 6, 10, 1, 17, 15, 5, 44, 23, 15, 107, 42, 35, 1, 252, 94, 70, 7, 588, 233, 129, 28, 1376, 585, 237, 84, 1, 3245, 1441, 468, 210, 9, 7717, 3481, 1026, 466, 45, 18485, 8319, 2434, 968, 165, 1, 44535, 19835, 5972, 1984, 495, 11, 107796, 47436

Offset: 0

Emeric Deutsch, Sep 13 2004

Row sums are the RNA secondary structure numbers (A004148).
T(n,0)=A190159(n).
Row n has 1+floor(n/3) terms.
Sum(k*T(n,k),k>=0) = A187260.

			Triangle starts:
1;
1;
1;
1,1;
1,3;
2,6;
6,10,1;
17,15,5;
44,23,15;
107,42,35,1;
T(6,2)=1 because we have (uhd)(uhd) (the two pertinent subwords are shown between parentheses).

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.
M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire, Sem. Loth. Comb. B08l (1984) 79-86.

Cf. A004148, A190159, A187260.

G.f.: G=G(t, z) satisfies aG^2 + bG + c = 0, where a=z^2*(1-z-z^2+2z^3-tz+2tz^2-2tz^3-tz^4+t^2z^4), b=-(1-z)(1-2z+2z^2+z^3-2tz^3), c=(1-z)^2.

The g.f. H(t,z), counting peakless Motzkin paths by the number of UH^bD (b is fixed) starting at level 0 (marked by t) and by length (marked by z), satisfies the equation H=1+zH+z^2*H(g-1-z^b + tz^b), where g=1+zg+z^2*g(g-1).

A110318 Number of arcs covered by other arcs in all RNA secondary structures of size n+5 (i.e., with n+5 nodes).

Original entry on oeis.org

1, 5, 17, 53, 157, 448, 1250, 3434, 9326, 25114, 67196, 178895, 474398, 1254072, 3306738, 8701193, 22857026, 59958380, 157098360, 411214120, 1075491286, 2810892598, 7342205478, 19168694232, 50023584613, 130497101659, 340325126923, 887307420361

Offset: 0

Emeric Deutsch, Jul 19 2005

nonn

			a(0)=1 because in the 8 (=A004148(5)) RNA secondary structures of size 5, namely 1/2/3/4/5, 13/2/4/5, 14/2/3/5, 15/2/3/4, 1/24/3/5, 1/25/3/4, 1/2/35/4 and 15/24/3 we have altogether 1 arc covered by another arc: in 15/24/3 the arc 24 is covered by the arc 15.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
W. R. Schmitt and M. S. Waterman, Linear trees and RNA secondary structure, Discrete Appl. Math., 51, 317-323, 1994.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1978), 261-272.
M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumeration en biologie moléculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.

Cf. A004148, A110317.

Q:=sqrt(1-2*z-z^2-2*z^3+z^4): G:=2*(1-2*z-z^3-(1-z)*Q)/Q/z^5/(1-z+z^2+Q)^2: Gser:=series(G,z=0,38): 1,seq(coeff(Gser,z^n),n=1..30);

CoefficientList[Series[2 (1 - 2 x - x^3 - (1 - x) Sqrt[1 - 2 x - x^2 - 2 x^3 + x^4]) / (x^5 Sqrt[1 - 2 x - x^2 - 2 x^3 + x^4] (1 - x + x^2 + Sqrt[1 - 2 x - x^2 - 2 x^3 + x^4])^2), {x, 0, 33}], x] (* Vincenzo Librandi, Jun 13 2017 *)

G.f.: 2(1-2z-z^3-(1-z)Q)/(z^5*Q(1-z+z^2+Q)^2), where Q:=sqrt(1-2z-z^2-2z^3+z^4).

a(n) = Sum_{k>=0} k*A110317(n+5,k).

A110333 Triangle read by rows: T(n,k) (n,k>=0) = number of peakless Motzkin paths of length n having k valleys (i.e., (1,-1) followed by (1,1)) at level zero (can be easily translated into RNA secondary structure terminology).

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 1, 33, 4, 70, 12, 152, 32, 1, 336, 82, 5, 754, 206, 18, 1714, 512, 56, 1, 3940, 1264, 163, 6, 9145, 3109, 456, 25, 21406, 7634, 1243, 88, 1, 50478, 18737, 3326, 284, 7, 119814, 46006, 8781, 868, 33, 286045, 113062, 22955, 2556, 129, 1, 686456

Offset: 0

Emeric Deutsch, Jul 20 2005

Row n (n >= 3) has floor(n/3) terms.

Row sums yield the RNA secondary structure numbers (A004148).

			T(10,2)=5 because we have HUH(DU)H(DU)HD, UH(DU)H(DU)HDH, UHH(DU)H(DU)HD, UH(DU)HH(DU)HD and UH(DU)H(DU)HHD, where U=(1,1), H=(1,0), D=(1,-1) and the valleys at level zero are shown between parentheses.
Triangle begins:
    1;
    1;
    1;
    2;
    4;
    8;
   16,  1;
   33,  4;
   70, 12;
  152, 32,  1;
  336, 62,  5;

W. R. Schmitt and M. S. Waterman, Linear trees and RNA secondary structure, Discrete Appl. Math., 51, 317-323, 1994.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1978), 261-272.
M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumeration en biologie moléculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.

Cf. A004148, A110334, A110335.

g:=(1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))/2/z^2: G:=(1+z^2*g-z^2*g*t-z^2+t*z^2)/(1-z-z^3*g-t*z^2*g+t*z^3*g+z^3+t*z^2-t*z^3): Gser:=simplify(series(G,z=0,23)): P[0]:=1: for n from 1 to 20 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 2 do print(1) od: for n from 3 to 20 do seq(coeff(t*P[n],t^k),k=1..floor(n/3)) od; # yields sequence in triangular form

T(n,0) = A110334(n).
Sum_{k>=0} k*T(n,k) = A110335(n-6) for n >= 6, 0 otherwise.
G.f.: (1 + z^2*g - tz^2*g - z^2 + tz^2)/(1 - z - z^3*g - tz^2*g + tz^3*g + z^3 + tz^2 - tz^3), where g = 1 + zg + z^2*g(g-1) = (1 - z + z^2 - sqrt(1 - 2z - z^2 - 2z^3 + z^4))/(2z^2) is the g.f. of the RNA secondary structure numbers (A004148).

Keyword tabf added by Michel Marcus, Apr 09 2013

A166285 Triangle read by rows: T(n,k) is the number of Dyck paths with no UUU's and no DDD's, of semilength n having k peak plateaux (0 <= k <= floor(n/3); U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 1, 2, 3, 1, 5, 3, 9, 8, 17, 19, 1, 34, 43, 5, 70, 97, 18, 147, 219, 56, 1, 313, 498, 160, 7, 673, 1140, 438, 32, 1459, 2623, 1168, 122, 1, 3185, 6061, 3062, 418, 9, 6995, 14053, 7932, 1342, 50, 15445, 32677, 20360, 4124, 225, 1, 34265, 76171, 51886, 12274, 895

Offset: 0

Emeric Deutsch, Oct 12 2009

A peak plateau is a run of consecutive peaks that is preceded by an upstep U and followed by a down step D; a peak consists of an upstep followed by a downstep.
Row n contains 1 + floor(n/3) entries.
Sum of entries in row n = A004148(n+1) (the secondary structure numbers).
T(n,0) = A166286(n).
Sum_{k=0..floor(n/3)} k*T(n,k) = A166287(n).

			T(4,1)=3 because we have UD(UUDUDD), (UUDUDD)UD, and (UUDUDUDD) (the peak plateaux are shown between parentheses).
Triangle starts:
   1;
   1;
   2;
   3,  1;
   5,  3;
   9,  8;
  17, 19,  1;
  34, 43,  5;

Cf. A004148, A166286, A166287.

F := RootOf(G = 1+z*G+z^2*G+z^3*G*((t-1)/(1-z)+G), G): Fser := series(F, z = 0, 20): for n from 0 to 17 do P[n] := sort(coeff(Fser, 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

G.f. G=G(t,z) satisfies G = 1 + zG + z^2*G + z^3*G[G+(t-1)/(1-z)].

A166295 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 UUDUDD's starting at level 0 (0 <= k <= floor(n/3); U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 1, 2, 3, 1, 6, 2, 12, 5, 26, 10, 1, 57, 22, 3, 128, 48, 9, 291, 109, 22, 1, 670, 250, 54, 4, 1558, 582, 129, 14, 3655, 1366, 311, 40, 1, 8639, 3232, 750, 109, 5, 20554, 7696, 1818, 284, 20, 49185, 18432, 4419, 730, 65, 1, 118301, 44368, 10776, 1856, 195, 6

Offset: 0

Emeric Deutsch, Oct 29 2009

Row n has 1 + floor(n/3) terms.
Sum of entries in row n = A004148(n+1) (the secondary structure numbers).
T(n,0) = A166296(n).
Sum_{k=0..floor(n/3)} k*T(n,k) = A166297(n).

			T(4,1)=2 because we have UDUUDUDD and UUDUDDUD.
Triangle starts:
   1;
   1;
   2;
   3,  1;
   6,  2;
  12,  5;
  26, 10,  1;

Cf. A004148, A166296, A166297.

G := 2/(1-z-z^2+2*z^3-2*t*z^3+sqrt(1-2*z-z^2-2*z^3+z^4)): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, j), j = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form

G.f.: G(t,z) = 1/(1-z-z^2+z^3-t*z^3-z^3*g), where g = 1+zg + z^2*g + z^3*g^2.

A166299 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n, having no ascents and no descents of length 1, and having k UUDD's starting at level 0.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 2, 0, 5, 2, 0, 1, 10, 4, 3, 0, 22, 11, 3, 0, 1, 50, 22, 6, 4, 0, 113, 49, 18, 4, 0, 1, 260, 114, 36, 8, 5, 0, 605, 260, 81, 26, 5, 0, 1, 1418, 604, 193, 52, 10, 6, 0, 3350, 1419, 444, 118, 35, 6, 0, 1, 7967, 3350, 1041, 288, 70, 12, 7, 0, 19055, 7966

Offset: 0

Emeric Deutsch, Nov 07 2009

Sum of entries in row n is the secondary structure number A004148(n-1) (n>=2).
Number of entries in row n is 1 + floor(n/2).
T(n,0)=A166300(n).
Sum(k*T(n,k), k>=0)=A075125(n+2).

			T(7,2)=3 because we have (UUDD)(UUDD)UUUDDD, (UUDD)UUUDDD(UUDD), and UUUDDD(UUDD)(UUDD) (the UUDD's starting at level 0 are shown between parentheses).
Triangle starts:
1;
0;
0,1;
1,0;
1,0,1;
2,2,0;
5,2,0,1;
10,4,3,0;

Cf. A004148, A166300, A075125

G := 2/(1+z+z^2+sqrt((1+z+z^2)*(1-3*z+z^2))-2*t*z^2): Gser := simplify(series(G, z = 0, 18)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

G.f.=G(t,z)=1/(1 + z - zg - tz^2), where g=g(z) satisfies g=1 + zg(g - 1 + z).
G.f. of column k is z^{2k}/(1 + z - zg)^{k+1} (k>=0).
G(t,z)=2/[1+z+z^2+sqrt((1+z+z^2)(1-3z+z^2)-2tz^2)].

A190159 Number of peakless Motzkin paths of length n and having no uhh...hd's starting at level 0, where u = (1, 1), h = (1, 0) and d = (1, -1).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 6, 17, 44, 107, 252, 588, 1376, 3245, 7717, 18485, 44535, 107796, 261937, 638673, 1562105, 3831655, 9423580, 23233536, 57412612, 142174255, 352770105, 876922947, 2183621209, 5446177428, 13603846132, 34028890577, 85234251090, 213760737693, 536733871490, 1349210120198

Offset: 0

Emeric Deutsch, May 05 2011

nonn

Can be easily expressed using RNA secondary structure terminology.

			a(5)=2 because among the 8 (=A004148(5)) peakless Motzkin paths of length 5 only hhhhh and uuhdd have no subword of type uhh...hd starting at level 0.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A098071, A004148

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(((1 -x)/(2*x^2*(1 -x -x^2 +2*x^3)))*(1 -2*x +2*x^2 +x^3 -Sqrt((1-2*x+2*x^2 + x^3)^2 -4*x^2*(1 -x -x^2 +2*x^3))))); // G. C. Greubel, Oct 22 2018

eq := z^2*(-z^2+2*z^3-z+1)*G^2-(1-z)*(z^3+2*z^2-2*z+1)*G+(1-z)^2 = 0: g := RootOf(eq,G): Gser := series(g,z=0,40): seq(coeff(Gser,z,n), n=0..35);

With[{a = x^2*(1 - x - x^2 + 2*x^3)}, CoefficientList[Series[((1 - x)/(2*a))*(1 - 2*x + 2*x^2 + x^3 - Sqrt[(1 - 2*x + 2*x^2 + x^3)^2 - 4*a]), {x, 0, 40}], x]] (* G. C. Greubel, Oct 22 2018 *)

x='x+O('x^40); Vec(((1-x)/(2*x^2*(1-x-x^2+2*x^3)))*(1-2*x+2*x^2 + x^3 - sqrt((1-2*x+2*x^2+x^3)^2 -4*x^2*(1-x-x^2+2*x^3)))) \\ G. C. Greubel, Oct 22 2018

a(n) = A098071(n,0).
G.f.=G=G(z) satisfies aG^2 + bG + c = 0, where a=z^2*(1-z-z^2+2z^3), b=-(1-z)(1-2z+2z^2+z^3), c=(1-z)^2.
G.f.: ((1 - x)/(2*x^2*(1 - x - x^2 + 2*x^3)))*(1 - 2*x + 2*x^2 + x^3 - sqrt((1 - 2*x + 2*x^2 + x^3)^2 - 4*x^2*(1 - x - x^2 + 2*x^3))). - G. C. Greubel, Oct 22 2018
D-finite with recurrence -(n+5)*(n+3)*(n-9)*a(n) +(-n^3-17*n^2-15*n+96)*a(n-1) +3*(7*n^3-43*n^2-20*n-40)*a(n-2) +(34*n^3-214*n^2+357*n-450)*a(n-3) -2*(n-3)*(8*n^2-48*n+183)*a(n-4) +(34*n^3-398*n^2+1461*n-1332)*a(n-5) +3*(7*n^3-83*n^2+220*n+196)*a(n-6) +(-n^3+35*n^2-327*n+822)*a(n-7) -(n-11)*(n+3)*(n-9)*a(n-8)=0. - R. J. Mathar, Jul 24 2022

cp's OEIS Frontend

A097860 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n with k peaks (n>=0, 0<=k<=floor(n/2)).

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A098056 Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n containing k subwords of the type uh^ju, dH^jd, or dh^ju for some j>0, where u=(1,1), d=(1,-1) and h=(1,0) (can be easily expressed using RNA secondary structure terminology).

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A098057 Number of peakless Motzkin paths with no U H^j U, no D H^j D and no D H^jU (j>0), where U=(1,1), D=(1,-1) and H=(1,0) (can be easily expressed using RNA secondary structure terminology).

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A098071 Triangle read by rows: T(n,k) is number of peakless Motzkin paths of length n and having k uhh...hd's starting at level 0, where u=(1,1), h=(1,0) and d=(1,-1) (can be easily expressed using RNA secondary structure terminology).

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A110318 Number of arcs covered by other arcs in all RNA secondary structures of size n+5 (i.e., with n+5 nodes).

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A110333 Triangle read by rows: T(n,k) (n,k>=0) = number of peakless Motzkin paths of length n having k valleys (i.e., (1,-1) followed by (1,1)) at level zero (can be easily translated into RNA secondary structure terminology).

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A166285 Triangle read by rows: T(n,k) is the number of Dyck paths with no UUU's and no DDD's, of semilength n having k peak plateaux (0 <= k <= floor(n/3); U=(1,1), D=(1,-1)).

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A166295 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 UUDUDD's starting at level 0 (0 <= k <= floor(n/3); U=(1,1), D=(1,-1)).

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