A004148 -id:A004148 - cp's OEIS frontend

A162984 Number of Dyck paths with no UUU's and no DDD's of semilength n and having k UUDUDD's (0<=k<=floor(n/3); U=(1,1), D=(1,-1)).

1, 1, 2, 3, 1, 6, 2, 12, 5, 25, 11, 1, 53, 26, 3, 114, 62, 9, 249, 148, 25, 1, 550, 355, 69, 4, 1227, 853, 189, 14, 2760, 2057, 509, 46, 1, 6253, 4973, 1359, 145, 5, 14256, 12050, 3600, 446, 20, 32682, 29256, 9484, 1334, 75, 1, 75293, 71154, 24870, 3914, 265, 6

Offset: 0

Emeric Deutsch, Oct 11 2009

T(n,k) is the number of weighted lattice paths in B(n) having k peaks. The members of B(n) are paths of weight n that start at (0,0), end on but never go below the horizontal axis, and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps. A peak is a (1,1)-step followed by a (1,-1)-step. Example: T(7,2)=3. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and U=(1,1), D=(1,-1), we have hUDUD, UDhUD, and UDUDh.

Number of entries in row n is 1+floor(n/3).

			T(4,1) = 2 because we have UDUUDUDD and UUDUDDUD.
Triangle starts:
1;
1;
2;
3,   1;
6,   2;
12,  5;
25, 11, 1;
53, 26, 3;

Alois P. Heinz, Rows n = 0..250, 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, A110320, A162985, A246177.

G := ((1-z-z^2+z^3-t*z^3-sqrt(1-2*z-z^2-2*t*z^3-z^4-2*z^5+z^6+2*t*z^4+2*t*z^5-2*t*z^6+t^2*z^6))*1/2)/z^3: 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
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y<0 or y>x or t=9, 0,
     `if`(x=0, 1, expand(b(x-1, y+1, [2, 3, 9, 5, 3, 2, 2, 2][t])+
     `if`(t=6, z, 1) *b(x-1, y-1, [8, 8, 4, 7, 6, 7, 9, 7][t]))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jun 10 2014

b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x || t == 9, 0, If[x == 0, 1, Expand[b[x-1, y+1, {2, 3, 9, 5, 3, 2, 2, 2}[[t]] ] + If[t == 6, z, 1]*b[x-1, y-1, {8, 8, 4, 7, 6, 7, 9, 7}[[t]] ]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 1]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)

G.f.: G=G(t,z) satisfies G = 1 + zG + z^2*G + z^3*(G-1+t)G.
Sum of entries in row n = A004148(n+1).
T(n,0) = A162985(n).
Sum(k*T(n,k), k=0..floor(n/3)) = A110320(n-2).

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

Emeric Deutsch, Oct 12 2009

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).

			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.

Cf. A004148, A000045, A166292, A166293, A166294

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

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

Emeric Deutsch, Oct 12 2009

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).

			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.

Cf. A004148, A166291, A166292, A166294

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

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).

A166297 Number of UUDUDD's starting at level 0 in all Dyck paths of semilength n with no UUU's and no DDD's (U=(1,1), D=(1,-1)).

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 12, 28, 66, 156, 370, 882, 2112, 5079, 12264, 29725, 72298, 176414, 431754, 1059595, 2607090, 6429913, 15893330, 39365876, 97692372, 242875105, 604836072, 1508619585, 3768496102, 9426815859, 23612178180, 59217406914

Offset: 0

Emeric Deutsch, Oct 29 2009

nonn

a(n) = Sum_{k=0..floor(n/3)} k*A166295(n,k).

			a(3)=1 because in UDUDUD, UDUUDD, UUDDUD, and UUDUDD we have 0+0+0+1=1 UUDUDD's starting at level 0.

Cf. A166295.

Cf. A004148. - Emeric Deutsch, Nov 10 2009

G := 4*z^3/(1-z-z^2+sqrt(1-2*z-z^2-2*z^3+z^4))^2: Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 32);

CoefficientList[Series[4*x^3/(1-x-x^2+Sqrt[1-2*x-x^2-2*x^3+x^4])^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

a(n):=2*sum((binomial(n-1-k,k)*binomial(n-1-k,k+2))/(n-1-k),k,0,(n-2)/2); /* Vladimir Kruchinin, Oct 13 2020 */

G.f.: G(z) = 4*z^3/(1-z-z^2+sqrt(1-2*z-z^2-2*z^3+z^4))^2.
a(n) = A004148(n+1) - A004148(n) - A004148(n-1) for n>=3. - Emeric Deutsch, Nov 10 2009
a(n) ~ sqrt(5 + 3*sqrt(5)) * ((3+sqrt(5))/2)^n / (sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 20 2014
a(n) = 2*Sum_{k=0..(n-2)/2} C(n-k-1,k)*C(n-k-1,k+2)/(n-k-1). - Vladimir Kruchinin, Oct 13 2020
D-finite with recurrence +(n+3)*a(n) +(-3*n-4)*a(n-1) +(n-4)*a(n-3) +2*(2*n-7)*a(n-4) +(n-6)*a(n-5) +(-n+7)*a(n-6)=0. - R. J. Mathar, Jul 24 2022

A167634 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 peaks at odd level.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 3, 1, 5, 2, 1, 1, 10, 5, 1, 14, 12, 9, 2, 7, 38, 27, 9, 1, 43, 60, 57, 22, 3, 36, 156, 146, 69, 15, 1, 143, 284, 326, 176, 45, 4, 166, 672, 784, 482, 155, 23, 1, 509, 1320, 1780, 1224, 453, 82, 5, 731, 2981, 4162, 3160, 1354, 313, 33, 1, 1915, 6104

Offset: 0

Emeric Deutsch, Nov 08 2009

Sum of entries in row n is the secondary structure number A004148(n-1) (n >= 2).
Row n contains ceiling(n/2) entries (n >= 1).
T(n,0) = A167635(n).
Sum_{k>=0} k*T(n,k) = A167636(n).

			T(5,1)=3 because we have UUDDUU(UD)DD, UU(UD)DDUUDD, and UUUU(UD)DDDD (the odd-level peaks are shown between parentheses).
Triangle starts:
   1;
   0;
   1;
   0,  1;
   2,  0;
   0,  3,  1;
   5,  2,  1;
   1, 10,  5,  1;
  14, 12,  9,  2;

Cf. A004148, A167635, A167636, A167637.

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

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

The trivariate g.f. G=G(t,s,z), where t marks odd-level peaks, s marks even-level peaks, and z marks semilength, satisfies aG^2 - bG + c = 0, where a = z(1+z-sz^2), b=(1+z-tz^2)(1+z-sz^2), c=1+z-tz^2.

A167637 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 peaks at even level.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 0, 1, 3, 3, 1, 5, 8, 4, 0, 5, 13, 12, 6, 1, 15, 32, 27, 8, 0, 21, 59, 61, 33, 10, 1, 51, 134, 147, 76, 15, 0, 85, 267, 327, 208, 75, 15, 1, 188, 584, 771, 528, 186, 26, 0, 344, 1209, 1734, 1329, 585, 150, 21, 1, 730, 2608, 4008, 3344, 1595, 408, 42, 0

Offset: 0

Emeric Deutsch, Nov 08 2009

Sum of entries in row n is the secondary structure number A004148(n-1) (n>=2).
Row n contains 1 + floor(n/2) entries.
T(n,0) = A167638(n).
Sum_{k>=0} k*T(n,k) = A167639(n).

			T(6,2)=3 because we have U(UD)DUUU(UD)DDD, UUU(UD)DDDU(UD)D, and UUU(UD)DU(UD)DDD (the even-level peaks are shown between parentheses).
Triangle starts:
  1;
  0;
  0,  1;
  1,  0;
  0,  1,  1;
  2,  2,  0;
  1,  3,  3,  1;
  5,  8,  4,  0;
  5, 13, 12,  6,  1;
  ...

Cf. A004148, A167634, A167638, A167639.

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

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

The trivariate g.f. G=G(t,s,z), where t marks odd-level peaks, s marks even-level peaks, and z marks semilength, satisfies aG^2 - bG + c = 0, where a = z(1+z-sz^2), b=(1+z-tz^2)(1+z-sz^2), c=1+z-tz^2.

A182896 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,-1)-returns to the horizontal axis. The members of L_n are paths of weight n that start at (0,0), end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

Original entry on oeis.org

1, 1, 2, 4, 1, 8, 3, 17, 9, 37, 25, 1, 82, 66, 5, 185, 171, 20, 423, 437, 70, 1, 978, 1107, 225, 7, 2283, 2790, 686, 35, 5373, 7009, 2015, 147, 1, 12735, 17574, 5760, 553, 9, 30372, 44019, 16135, 1932, 54, 72832, 110210, 44500, 6398, 264, 1, 175502, 275925, 121247, 20350, 1134, 11

Offset: 0

Emeric Deutsch, Dec 12 2010

			T(3,1)=1. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; exactly one of them, namely ud, has one (1,-1)-return to the horizontal axis.
Triangle starts:
   1;
   1;
   2;
   4,  1;
   8,  3;
  17,  9;
  37, 25, 1;
  82, 66, 5;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1750
M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

Cf. A051286, A004148, A182897.

eq := c = 1+z*c+z^2*c+z^3*c^2: c := RootOf(eq, c): G := 1/(1-z-z^2-t*z^3*c-z^3*c): Gser := simplify(series(G, z = 0, 19)): 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((1/3)*n)) end do: #yields sequence in triangular form

T(n)={[Vecrev(p) | p<-Vec(1/(1-x-x^2 - (1+y)*(1-x-x^2 - sqrt(1+x^4-2*x^3-x^2-2*x+O(x*x^n)))/2))]}
{ my(A=T(12)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Nov 05 2019

G.f.: G(t,z) = 1/(1-z-z^2-(1+t)z^3*c), where c satisfies c = 1 + zc + z^2*c + z^3*c^2.
Sum of entries in row n is A051286(n).
T(n,0) = A004148(n+1) (the secondary structure numbers).
Sum_{k=0..n} k*T(n,k) = A182897(n).

Data corrected by Andrew Howroyd, Nov 05 2019

A182900 Triangle read by rows: T(n,k) is the number of weighted lattice paths in B(n) having k valleys. The members of B(n) are paths of weight n that start at (0,0), end on but never go below the horizontal axis, and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps. A valley is a (1,-1)-step followed by a (1,1)-step.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 36, 1, 78, 4, 171, 14, 379, 43, 1, 848, 125, 5, 1912, 351, 20, 4341, 960, 71, 1, 9915, 2579, 235, 6, 22767, 6833, 745, 27, 52526, 17916, 2281, 108, 1, 121698, 46593, 6805, 399, 7, 283043, 120385, 19885, 1400, 35, 660579, 309416, 57141, 4712, 155, 1

Offset: 0

Emeric Deutsch, Dec 15 2010

Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
T(n,0)=A182901(n).
Sum(k*T(n,k), k>=0) = A182902(n).
For the distribution of the statistic "number of peaks" see A162984. A peak is a (1,1)-step followed by a (1,-1)-step.

			T(7,1)=4. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and U=(1,1), D=(1,-1), we have hUDUD, UDUDh, UDUhD, and UhDUD.
Triangle starts:
1;
1;
2;
4;
8;
17;
36,1;
78,4;
171,14;

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, A182901, A182902, A162984.

eq := z^3*(z+z^2+v-v*z-v*z^2)*F^2-(1-z-z^2-z^3+v*z^3)*F+1 = 0: F := RootOf(eq, F): Fser := simplify(series(F, z = 0, 20)): for n from 0 to 18 do P[n] := sort(coeff(Fser, z, n)) end do: 1; 1; 2; for n from 0 to 18 do seq(coeff(P[n], v, k), k = 0 .. floor((1/3)*n)-1) end do; # yields sequence in triangular form

G. f.: F=F(v,z) satisfies z^3*(z+z^2+v-vz-vz^2)F^2 - (1-z-z^2-z^3+vz^3)F+1=0 (z marks weight, v marks number of valleys).

The trivariate g.f. H(u,v,z), where u (v) marks peaks (valleys) and z marks weight is given by H=1+zH+z^2*H+z^3*(u-1+H)[v(H-1-zH-z^2*H)+1+zH+z^2*H].

A187256 Number of peakless Motzkin paths of length n, assuming that the (1,0)-steps come in 2 colors.

Original entry on oeis.org

1, 2, 4, 10, 28, 82, 248, 770, 2440, 7858, 25644, 84618, 281844, 946338, 3199728, 10885122, 37230352, 127951714, 441633812, 1530242954, 5320853260, 18560408050, 64932101224, 227767796482, 800928670232, 2822814469394, 9969770245948, 35280714655498

Offset: 0

Emeric Deutsch, May 03 2011

nonn

Ordinary peakless Motzkin paths are counted by A004148.

Also the number of Catalan words of length n avoiding the consecutive pattern 010. - Sela Fried, May 21 2025

			a(4)=28 because, denoting U=(1,1), D=(1,-1), and H=(1,0), we have 2^4=16 paths of shape HHHH, 2^2=4 paths of shape HUHD, 2^2 = 4 paths of shape UHDH, and 4 paths of shape UHHD.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv:1107.5490 [math.CO], 2011.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.

Cf. A004148, A328504

Column k=0 of A114848 (shifted). - Alois P. Heinz, Mar 31 2016

eq := G = 1+2*z*G+z^2*G*(G-1): G := RootOf(eq, G): Gser := series(G, z = 0, 30): seq(coeff(Gser, z, n), n = 0 .. 27);

CoefficientList[Series[(1 + (x-2)*x - Sqrt[(1 + (x-4)*x)*(1+x^2)])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 02 2014 *)
a[n_] := 2^n HypergeometricPFQ[{-n/2, (1 - n)/2, (1 - n)/2, 1 - n/2}, {2, -n, -n + 1}, 4]; Array[a, 28, 0] (* Peter Luschny, Jan 25 2020 *)

a(n):=sum(((-1)^i*binomial(n-i,i)*binomial(2*n-4*i+2,n-2*i))/(n-2*i+1),i,0,(n)/2); /* Vladimir Kruchinin, Jun 01 2014 */

my(x='x+O('x^50)); Vec((1 + (x-2)*x - sqrt((1 + (x-4)*x)*(1+x^2))) /( 2*x^2)) \\ G. C. Greubel, Feb 12 2017

G.f.: G(z) satisfies the equation G = 1 + 2*z*G + z^2*G*(G-1).
Conjecture: (n+2)*a(n) -2*(2*n+1)*a(n-1) +2*(n-1)*a(n-2) +2*(5-2*n)*a(n-3) +(n-4)*a(n-4) = 0. - R. J. Mathar, Nov 16 2011
a(n) = Sum_{i=0..n/2} (-1)^i*binomial(n-i,i)*binomial(2*n-4*i+2,n-2*i)/(n-2*i+1). - Vladimir Kruchinin, Jun 01 2014
a(n) ~ sqrt(24+14*sqrt(3)) * (2+sqrt(3))^n / (2*sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 02 2014
a(n) = 2^n*hypergeom([-n/2, (1 - n)/2, (1 - n)/2, 1 - n/2], [2, -n, -n + 1], 4). - Peter Luschny, Jan 25 2020

A190164 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 0,2,4,... .

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 0, 0, 1, 1, 3, 3, 0, 0, 1, 2, 4, 6, 4, 0, 0, 1, 4, 8, 9, 10, 5, 0, 0, 1, 7, 18, 19, 16, 15, 6, 0, 0, 1, 12, 35, 48, 36, 25, 21, 7, 0, 0, 1, 22, 66, 102, 100, 60, 36, 28, 8, 0, 0, 1, 41, 132, 209, 229, 180, 92, 49, 36, 9, 0, 0, 1, 76, 266, 450, 504, 440, 294, 133, 64, 45, 10, 0, 0, 1

Offset: 0

Emeric Deutsch, May 06 2011

Sum of entries in row n is A004148(n) (the RNA secondary structure numbers).
T(n,0)=A190165(n).
Sum_{k>=0} k*T(n,k) = A190166(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 the equation
z^2*(1-tz+z^2)*H^2 - (1-tz+z^2)*(1-sz+z^2)*H + 1-sz+z^2 = 0.

			T(5,2)=3 because we have h'h'uhd, h'uhdh', and uhdh'h', where u=(1,1), h=(1,0), d=(1,-1) (the even-level h-steps are marked).
Triangle starts:
  1;
  0, 1;
  0, 0, 1;
  1, 0, 0, 1;
  1, 2, 0, 0, 1;
  1, 3, 3, 0, 0, 1;

Cf. A004148, A190165, A190166, A110236, A190167.

eq := z^2*(1-t*z+z^2)*G^2-(1-z+z^2)*(1-t*z+z^2)*G+1-z+z^2 = 0: g := RootOf(eq, G): Gser := simplify(series(g, z = 0, 15)): for n from 0 to 13 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form

m = 13; G[_] = 0;
Do[G[z_] = -((z^2 G[z]^2 (-t z + z^2 + 1) + z^2 - z + 1)/((z^2 - z + 1)(t z - z^2 - 1))) + O[z]^m, {m}];
CoefficientList[#, t]& /@ CoefficientList[G[z], z] // Flatten (* Jean-François Alcover, Nov 15 2019 *)

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

cp's OEIS Frontend

A162984 Number of Dyck paths with no UUU's and no DDD's of semilength n and having k UUDUDD's (0<=k<=floor(n/3); U=(1,1), D=(1,-1)).

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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)).

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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)).

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A166297 Number of UUDUDD's starting at level 0 in all Dyck paths of semilength n with no UUU's and no DDD's (U=(1,1), D=(1,-1)).

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A167634 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 peaks at odd level.

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A167637 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 peaks at even level.

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