A004148 -id:A004148 - cp's OEIS frontend

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

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

Emeric Deutsch, May 06 2011

nonn

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

a(n)=A110236(n) - A190169(n).

			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).

Cf. A190164, A110236, A190169, A004148.

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);

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

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

Emeric Deutsch, May 06 2011

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).

			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;

Cf. A004148, A190171, A089735.

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

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).

A190172 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n having k UHD's; here U=(1,1), H=(1,0), and D=(1,-1).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 4, 4, 8, 8, 1, 16, 18, 3, 33, 40, 9, 69, 90, 25, 1, 146, 204, 69, 4, 312, 467, 183, 16, 673, 1074, 479, 56, 1, 1463, 2481, 1239, 185, 5, 3202, 5752, 3180, 576, 25, 7050, 13378, 8104, 1734, 105, 1, 15605, 31196, 20544, 5076, 405, 6, 34705, 72912, 51852, 14546, 1451, 36

Offset: 0

Emeric Deutsch, May 06 2011

Number of entries in row n is 1+floor(n/3).
Sum of entries in row n = A004148 (the RNA secondary structure numbers).
T(n,0)=A004149(n).
Sum(k*T(n,k),k>=0)=A110236(n-2) (n>=3).

			T(5,1)=4 because we have HHUHD, HUHDH, UHDH, and UUHDD.
Triangle starts:
1;
1;
1;
1,1;
2,2;
4,4;
8,8,1;
16,18,3;

Cf. A004148, A004149, A110236

eq := G = 1+z*G+z^2*G*(G-1-z+t*z): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 25)): 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, k), k = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form

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

A203019 Number of elevated peakless Motzkin paths.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 4, 8, 17, 37, 82, 185, 423, 978, 2283, 5373, 12735, 30372, 72832, 175502, 424748, 1032004, 2516347, 6155441, 15101701, 37150472, 91618049, 226460893, 560954047, 1392251012, 3461824644, 8622571758, 21511212261, 53745962199

Offset: 0

Panayotis Vlamos and Antonios Panayotopoulos, Dec 27 2011

nonn

Essentially the same as A004148: a(0)=a(1)=0 and a(n) = A004148(n-2) for n>=2.

			G.f. = x^2 + x^3 + x^4 + 2*x^5 + 4*x^6 + 8*x^7 + 17*x^8 + 37*x^9 + ...

A. Panayotopoulos and P. Tsikouras, Properties of meanders, JCMCC 46 (2003), 181-190.
A. Panayotopoulos and P. Vlamos, Meandric Polygons, Ars Combinatoria 87 (2008), 147-159.

Muniru A Asiru, Table of n, a(n) for n = 0..300
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
I. Jensen, Enumeration of plane meanders, arXiv:cond-mat/9910313 [cond-mat.stat-mech], 1999.
S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Theoretical Computer Science Vol. 117 (1993) p. 232.
A. Panayotopoulos and P. Tsikouras, The multimatching property of nested sets, Math. & Sci. Hum. 149 (2000), 23-30.
A. Panayotopoulos and P. Tsikouras, Meanders and Motzkin Words, J. Integer Seqs., Vol. 7, 2004.
A. Panayotopoulos and P. Vlamos, Cutting Degree of Meanders, Artificial Intelligence Applications and Innovations, IFIP Advances in Information and Communication Technology, Volume 382, 2012, pp 480-489; DOI 10.1007/978-3-642-33412-2_49. - From _N. J. A. Sloane_, Dec 29 2012

List([0..40],n->Sum([0..Int((n-1)/2)],m->Binomial(2*m+1,m)*Sum([0..n-2*m-2],k->(Binomial(k,n-2*m-k-2)*Binomial(2*m+k,k)*(-1)^(n-k))/(2*m+1)))); # Muniru A Asiru, Aug 13 2018

terms = 34;
A[] = 0; Do[A[x] = x (x - A[x] / (A[x] - 1)) + O[x]^terms, {terms}];
CoefficientList[A[x], x] (* Jean-François Alcover, Jul 27 2018, after Michael Somos *)
Table[Sum[Binomial[2*m + 1, m]*Sum[(Binomial[k, n - 2*m - k - 2]* Binomial[2*m + k, k]*(-1)^(n - k))/(2*m + 1), {k, 0, n - 2*m - 2}], {m, 0, Floor[(n - 1)/2]}], {n, 0, 50}] (* G. C. Greubel, Aug 12 2018 *)

a(n):=sum((binomial(2*m+1,m)*sum(binomial(k,n-2*m-k-2)*binomial(2*m+k,k)*(-1)^(n-k),k,0,n-2*m-2))/(2*m+1),m,0,(n-1)/2); /* Vladimir Kruchinin, Mar 12 2016 */

{a(n) = local(A); A = O(x); for( k=1, ceil(n / 3), A = x^2 / (1 - x / (1 - A))); polcoeff( A, n)} /* Michael Somos, May 12 2012 */

G.f.: x^2 / (1 - x / (1 - x^2 / (1 - x / (1 - x^2 / (1 - x / (1 - x^2 / ...)))))). - Michael Somos, May 12 2012
G.f. A(x) =: y satisfies y / x = x + y / (1 - y). - Michael Somos, Jan 31 2014
G.f. A(x) =: y satisfies y = x^2 + (x - x^2)*y + y*y. - Michael Somos, Jan 31 2014
Given g.f. A(x), then B(x) = A(x)/x satisfies B(-B(-x)) = x. - Michael Somos, Jan 31 2014
a(n) = Sum_{m=0..(n-1)/2}((binomial(2*m+1,m)*Sum_{k=0..n-2*m-2}(binomial(k,n-2*m-k-2)*binomial(2*m+k,k)*(-1)^(n-k)))/(2*m+1)). - Vladimir Kruchinin, Mar 12 2016
a(n) ~ 5^(1/4) * phi^(2*n - 2) / (2*sqrt(Pi)*n^(3/2)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 14 2018
D-finite with recurrence n*a(n) +(-2*n+3)*a(n-1) +(-n+3)*a(n-2) +(-2*n+9)*a(n-3) +(n-6)*a(n-4)=0. - R. J. Mathar, Jan 25 2023

A247288 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n having k weak peaks.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 2, 1, 1, 0, 8, 4, 3, 1, 1, 0, 16, 8, 7, 4, 1, 1, 0, 32, 16, 17, 10, 5, 1, 1, 0, 64, 32, 41, 26, 14, 6, 1, 1, 0, 128, 64, 98, 66, 39, 19, 7, 1, 1, 0, 256, 128, 232, 164, 107, 56, 25, 8, 1, 1, 0, 512, 256, 544, 400, 286, 164, 78, 32, 9, 1

Offset: 0

Emeric Deutsch, Sep 14 2014

A weak peak of a Motzkin path is a vertex on the top of a hump.
A hump is an upstep followed by 0 or more flatsteps followed by a downstep. For example, the peakless Motzkin path uhu*h*ddu*h*h*d where u=(1,1), h=(1,0), d(1,-1), has 5 weak peaks (shown by the stars).
Row n (n>=1) contains n entries.
Sum of entries in row n is the RNA secondary structure number A004148(n).
Sum(k*T(n,k), 0<=k<=n) = A247289(n).

			Row 4 is 1,0,2,1 because the peakless Motzkin paths hhhh, u*h*dhh, hu*h*dh, and u*h*h*d  have 0, 2, 2, and 3 weak peaks (shown by the stars).
Triangle starts:
1;
1;
1,0;
1,0,1;
1,0,2,1;
1,0,4,2,1;
1,0,8,4,3,1;

Alois P. Heinz, Rows n = 0..141, flattened

Cf. A004148, A247289.

eq := G = 1+z*G+z^2*(G-1-z/(1-z)+t^2*z/(1-t*z))*G: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 16)): for n from 0 to 14 do P[n] := sort(expand(coeff(Gser, z, n))) end do: 1; for n to 14 do seq(coeff(P[n], t, k), k = 0 .. n-1) end do; # yields sequence in triangular form

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

A247289 Number of weak peaks in all peakless Motzkin paths of length n.

Original entry on oeis.org

0, 0, 0, 2, 7, 18, 45, 110, 267, 652, 1602, 3960, 9845, 24594, 61689, 155270, 391962, 991968, 2515964, 6393610, 16275174, 41491776, 105922244, 270734244, 692756227, 1774418286, 4549173861, 11672860634, 29975156134, 77029918152, 198083586300, 509692521982

Offset: 0

Emeric Deutsch, Sep 14 2014

nonn

A weak peak of a Motzkin path is a vertex on the top of a hump.
A hump is an upstep followed by 0 or more flatsteps followed by a downstep. For example, the peakless Motzkin path uhu*h*ddu*h*h*d, where u=(1,1), h=(1,0), d(1,-1), has 5 weak peaks (shown by the stars).
a(n) = Sum(k*A247288(n,k), 0<=k<=n-1).

			a(4)=7 because the peakless Motzkin paths u*h*dhh, hu*h*dh, and u*h*h*d  have 0, 2, 2, and 3 weak peaks (shown by the stars).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A004148, A247288.

f := (2-z)*z^3*g/((1-z)^2*(1-z+z^2-2*z^2*g)): eqg := g = 1+z*g+z^2*g*(g-1): g := RootOf(eqg, g): fser := series(f, z = 0, 35): seq(coeff(fser, z, n), n = 0 .. 33);

G.f.: (2-z)*z^3*g/((1-z)^2*(1-z+z^2-2*z^2*g)), where g is defined by g = 1 + z*g + z^2*g*(g-1).

D-finite with recurrence n*(n-1)*a(n) +(-7*n^2+28*n-31)*a(n-1) +(n-2)*(13*n-48)*a(n-2) +(-5*n^2+21*n-6)*a(n-3) +(7*n^2-43*n+82)*a(n-4) -(13*n-24)*(n-5)*a(n-5) +(4*n-5)*(n-6)*a(n-6)=0. - R. J. Mathar, Jul 24 2022

A247296 Number of uhd and uHd in all weighted lattice paths B(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 7, 18, 45, 112, 281, 706, 1778, 4490, 11363, 28814, 73199, 186257, 474635, 1211122, 3094171, 7913765, 20261142, 51921920, 133171656, 341836748, 878104607, 2257208148, 5805964495, 14942942127, 38480449261, 99145105834, 255573465001, 659114680270

Offset: 0

Emeric Deutsch, Sep 16 2014

nonn

B(n) is the set of lattice paths of weight n that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: h = (1,0) of weight 1, H = (1,0) of weight 2, u = (1,1) of weight 2, and d = (1,-1) of weight 1. The weight of a path is the sum of the weights of its steps.

a(n) = A110320(n-3) + A110320(n-4) (n>=5).

			a(6)=7 because among the 37 (=A004148(7)) members of B(6) only (uhd)hh, h(uhd)h, hh(uhd), H(uhd), (uhd)H, (uHd)h, and h(uHd) contain uhd or uHd (shown between parentheses).
G.f. = x^4 + 3*x^5 + 7*x^6 + 18*x^7 + 45*x^8 + 112*x^9 + 281*x^10 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.

Cf. A004148, A110320, A247294.

m:=30; R:=PowerSeriesRing(Integers(), m); [0,0,0,0] cat Coefficients(R!(x*(x+1)*(-1 +(1-x-x^2 )/sqrt((1-3*x+x^2)*(1+x+x^2)) )/2)); // G. C. Greubel, Aug 05 2018

eqg := g = 1+z*g+z^2*g+z^3*g^2: g := RootOf(eqg, g): H := z^4*(1+z)*g/(1-z-z^2-2*z^3*g): Hser := series(H, z = 0, 40): seq(coeff(Hser, z, n), n = 0 .. 35);

a[ n_] := With[{t = (1 - 3 x + x^2) (1 + x + x^2)}, SeriesCoefficient[ x (x + 1) (-1 + (1 - x - x^2) / Sqrt[t]) / 2, {x, 0, n}]]; (* Michael Somos, Sep 16 2014 *)

x='x+O('x^30); concat(vector(4), Vec(x*(x+1)*(-1 +(1-x-x^2 )/sqrt((1-3*x+x^2)*(1+x+x^2)))/2)) \\ G. C. Greubel, Aug 05 2018

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

D-finite with recurrence +(n-1)*(202*n-903)*a(n) +(-250*n^2+1095*n-691)*a(n-1) +(-510*n^2+4095*n-8039)*a(n-2) +(-558*n^2+4287*n-7831)*a(n-3) +(-106*n^2+1587*n-4575)*a(n-4) +(154*n-547)*(n-7)*a(n-5)=0. - R. J. Mathar, Jul 24 2022

A247299 Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having a total of k h- and H-steps at level 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 2, 1, 3, 1, 2, 4, 3, 3, 4, 1, 5, 6, 9, 5, 6, 5, 1, 10, 15, 15, 16, 9, 10, 6, 1, 22, 33, 33, 32, 26, 16, 15, 7, 1, 50, 71, 78, 66, 60, 41, 27, 21, 8, 1, 113, 163, 171, 158, 125, 103, 64, 43, 28, 9, 1, 260, 374, 391, 360, 295, 225, 167, 99, 65, 36, 10, 1

Offset: 0

Emeric Deutsch, Sep 17 2014

B(n) is the set of lattice paths of weight n that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: h = (1,0) of weight 1, H = (1,0) of weight 2, u = (1,1) of weight 2, and d = (1,-1) of weight 1. The weight of a path is the sum of the weights of its steps.
Row n contains n+1 entries.
Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
T(n,0) = A166300(n).
Sum(k*T(n,k), k=0..n) = A247300(n)

			Row 3 is 1,0,2,1 because B(3) = {ud, hH, Hh, hhh}.
Triangle starts:
1;
0,1;
0,1,1;
1,0,2,1;
1,2,1,3,1;
2,4,3,3,4,1;

Alois P. Heinz, Rows n = 0..140, flattened
M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.

Cf. A004148, A166300, A247300,

eqg := g = 1+z*g+z^2*g+z^3*g^2: g := RootOf(eqg, g): G := 1/(1-t*z-t*z^2-z^3*g): Gser := simplify(series(G, z = 0, 16)): for n from 0 to 13 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 13 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y) option remember; `if`(y<0 or y>n or n<0, 0,
      `if`(n=0, 1, expand(`if`(y=0, x, 1)*(b(n-1, y)+
      b(n-2, y)) +b(n-2, y+1) +b(n-1, y-1))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..14);  # Alois P. Heinz, Sep 17 2014

b[n_, y_] := b[n, y] = If[y<0 || y>n || n<0, 0, If[n == 0, 1, Expand[If[y == 0, x, 1]*(b[n-1, y] + b[n-2, y]) + b[n-2, y+1] + b[n-1, y-1]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

G.f. G = 1/(1 - t*z - t*z^2 - z^3*g), where g is given by g = 1 + z*g + z^2*g + z^3*g^2.

A273713 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k doublerises (n>=2, k>=0). A doublerise in a bargraph is any pair of adjacent up steps.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 8, 13, 9, 4, 1, 17, 32, 28, 14, 5, 1, 37, 80, 81, 50, 20, 6, 1, 82, 201, 231, 165, 80, 27, 7, 1, 185, 505, 653, 526, 295, 119, 35, 8, 1, 423, 1273, 1824, 1644, 1036, 483, 168, 44, 9, 1, 978, 3217, 5058, 5034, 3535, 1848, 742, 228, 54, 10, 1

Offset: 2

Emeric Deutsch, May 28 2016

Number of entries in row n is n-1.
Sum of entries in row n = A082582(n).
T(n,0) = A004148(n-1) (the 2ndary structure numbers).
T(n,1) = A110320(n-2).
Sum(k*T(n,k), k>=0) = A273714(n).

			Row 4 is 2,2,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and the corresponding drawings show that they have 0, 0, 1, 1, 2 doublerises.
Triangle starts
1;
1,1;
2,2,1;
4,5,3,1;
8,13,9,4,1

Alois P. Heinz, Rows n = 2..150, flattened
M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016

Cf. A004148, A082582, A110320, A273714.

eq := z*G^2-(1-z-t*z-z^2)*G+z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 22)): for n from 2 to 20 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 20 do seq(coeff(P[n], t, j), j = 0 .. n-2) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y, t) option remember; expand(`if`(n=0, (1-t),
      `if`(t<0, 0, b(n-1, y+1, 1)*`if`(t=1, z, 1))+
      `if`(t>0 or y<2, 0, b(n, y-1, -1))+
      `if`(y<1, 0, b(n-1, y, 0))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..n-2))(b(n, 0$2)):
seq(T(n), n=2..16);  # Alois P. Heinz, Jun 06 2016

b[n_, y_, t_] := b[n, y, t] = Expand[If[n == 0, 1 - t, If[t < 0, 0, b[n - 1, y + 1, 1]*If[t == 1, z, 1]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1]] + If[y < 1, 0, b[n - 1, y, 0]]]];
T[n_] := Function [p, Table[Coefficient[p, z, i], {i, 0, n - 2}]][b[n, 0, 0]];
Table[T[n], {n, 2, 16}] // Flatten (* Jean-François Alcover, Jul 29 2016, after Alois P. Heinz *)

G.f.: G = G(t,z) satisfies zG^2 - (1 - z - tz - z^2)G + z^2 = 0.

A292461 Expansion of (1 - x - x^2 + sqrt((1 - x - x^2)^2 - 4*x^3))/2 in powers of x.

Original entry on oeis.org

1, -1, -1, -1, -1, -2, -4, -8, -17, -37, -82, -185, -423, -978, -2283, -5373, -12735, -30372, -72832, -175502, -424748, -1032004, -2516347, -6155441, -15101701, -37150472, -91618049, -226460893, -560954047, -1392251012, -3461824644, -8622571758, -21511212261

Offset: 0

Seiichi Manyama, Sep 16 2017

sign

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

Cf. A004148, A203019, A292440, A292460.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x-x^2 +Sqrt((1-x-x^2)^2 -4*x^3))/2)); // G. C. Greubel, Aug 13 2018

CoefficientList[Series[(1-x-x^2 +Sqrt[(1-x-x^2)^2 -4*x^3])/2, {x, 0, 50} ], x] (* G. C. Greubel, Aug 13 2018 *)

x='x+O('x^50); Vec((1-x-x^2 +sqrt((1-x-x^2)^2 -4*x^3))/2) \\ G. C. Greubel, Aug 13 2018

Convolution inverse of A292460.
Let f(x) = (1 - x - x^2 - sqrt((1 - x - x^2)^2 - 4*x^3))/(2*x^3).
G.f.: 1-x-x^2-x^3/(1-x-x^2-x^3/(1-x-x^2-x^3/(1-x-x^2-x^3/(1-x-x^2-x^3/(... (continued fraction).
G.f.: 1/f(x) = 1 - x - x^2 - x^3*f(x).
a(n) = -A292460(n-3) for n > 2.
a(n) ~ -5^(1/4) * phi^(2*n - 2) / (2 * sqrt(Pi) * n^(3/2)), where phi is the golden ratio (1+sqrt(5))/2. - Vaclav Kotesovec, Sep 17 2017, simplified Dec 06 2021
Conjecture D-finite with recurrence: n*a(n) +(-2*n+3)*a(n-1) +(-n+3)*a(n-2) +(-2*n+9)*a(n-3) +(n-6)*a(n-4)=0. - R. J. Mathar, Jan 24 2020

cp's OEIS Frontend

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

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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).

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A190172 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n having k UHD's; here U=(1,1), H=(1,0), and D=(1,-1).

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A203019 Number of elevated peakless Motzkin paths.

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Maxima

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A247288 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n having k weak peaks.

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A247289 Number of weak peaks in all peakless Motzkin paths of length n.

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A247296 Number of uhd and uHd in all weighted lattice paths B(n).

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