Sergi Elizalde - cp's OEIS frontend

User: Sergi Elizalde

Sergi Elizalde has authored 21 sequences. Here are the ten most recent ones:

A376110 Number of permutations of {1,2,...,n} that are non-self-overlapping as Hertzsprung patterns; also called non-extendible.

1, 1, 0, 4, 18, 106, 658, 4778, 38770, 352458, 3546170, 39179282, 471653322, 6146403266, 86212578962, 1295136607114, 20747437026442, 353059209467042, 6360348815730370, 120931046165866362, 2420054522391186274, 50846927248165344442, 1119121906010637564906, 25749587951077654272898

Offset: 0

Sergi Elizalde, Sep 10 2024

nonn

Equivalently, a(n) is the number of permutations of {1,2,...,n} that have no proper Hertzsprung bifix (i.e., a prefix and a suffix of length i

			For n=4, the a(4) = 18 non-self-overlapping permutations of {1,2,3,4} are all but 1234, 4321, 1324, 4231, 2143, 3412.

Miklós Bóna, Where the monotone pattern (mostly) rules, Discrete Math. 308 (2008), no. 23, 5782-5788.
Anders Claesson, From Hertzsprung’s problem to pattern-rewriting systems, Algebraic Combinatorics 5 (2022), 1257-1277.
A. N. Myers, Counting permutations by their rigid patterns, J. Combin. Theory, Series A, Vol. 99, No. 2 (2002), pp. 345-357.

NonOverPerms:= proc(n) option remember;
  n!-add((i! + NonOverPerms(i))*(n-2*i)!, i=1..floor(n/2))
end:
seq(NonOverPerms(n), n=0..25);

a(n) = n! - Sum_{i=1..floor(n/2)} (i! + a(i))*(n-2*i)!.

A341445 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having degree of symmetry k (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 0, 2, 2, 0, 3, 2, 6, 0, 6, 8, 8, 16, 0, 10, 16, 32, 24, 40, 0, 20, 52, 84, 108, 60, 90, 0, 35, 134, 262, 294, 310, 150, 210, 0, 70, 432, 816, 1008, 880, 816, 336, 448, 0, 126, 1248, 2544, 3192, 3208, 2460, 2100, 784, 1008, 0, 252

Offset: 1

Sergi Elizalde, Feb 12 2021

The degree of symmetry of a Dyck path is defined as the number of steps in the first half that are mirror images of steps in the second half, with respect to the reflection along the vertical line through the midpoint of the path.

			For n=4 there are 6 Dyck paths with degree of symmetry equal to 2: uuuddudd, uuduuddd, uududdud, uuddudud, uduududd, ududuudd.
Triangle begins:
     1;
     0,    2;
     2,    0,    3;
     2,    6,    0,    6;
     8,    8,   16,    0,   10;
    16,   32,   24,   40,    0,   20;
    52,   84,  108,   60,   90,    0,  35;
   134,  262,  294,  310,  150,  210,   0,   70;
   432,  816, 1008,  880,  816,  336, 448,    0, 126;
  1248, 2544, 3192, 3208, 2460, 2100, 784, 1008,   0, 252;
  ...

Sergi Elizalde, The degree of symmetry of lattice paths, arXiv:2002.12874 [math.CO], 2020.
Sergi Elizalde, Measuring symmetry in lattice paths and partitions, Sem. Lothar. Combin. 84B.26, 12 pp (2020).

Equivalent to A298645 with rows reversed.
Row sums give A000108.
Main diagonal gives A001405.
Column k=1 gives A298647 (for n>2).
Second subdiagonal gives 2*A191522.

b:= proc(x, y, v) option remember; expand(
      `if`(min(y, v, x-max(y, v))<0, 0, `if`(x=0, 1, (l-> add(add(
      `if`(y=v+(j-i)/2, z, 1)*b(x-1, y+i, v+j), i=l), j=l))([-1, 1]))))
    end:
g:= proc(n) option remember; add(b(n, j$2), j=0..n) end:
T:= (n, k)-> coeff(g(n), z, k):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Feb 12 2021

b[x_, y_, v_] := b[x, y, v] = Expand[If[Min[y, v, x - Max[y, v]] < 0, 0, If[x == 0, 1, Function[l, Sum[Sum[If[y == v + (j - i)/2, z, 1]*b[x - 1, y + i, v + j], {i, l}], {j, l}]][{-1, 1}]]]];
g[n_] := g[n] = Sum[b[n, j, j], {j, 0, n}];
T[n_, k_] := Coefficient[g[n], z, k];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Feb 13 2021, after Alois P. Heinz *)

A339754 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k symmetric vertices (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 0, 2, 0, 2, 3, 0, 2, 6, 6, 0, 4, 12, 16, 10, 0, 8, 24, 40, 40, 20, 0, 20, 60, 104, 120, 90, 35, 0, 50, 150, 270, 350, 330, 210, 70, 0, 140, 420, 768, 1040, 1080, 840, 448, 126, 0, 392, 1176, 2184, 3080, 3468, 3108, 2128, 1008, 252

Offset: 1

Sergi Elizalde, Feb 12 2021

A symmetric vertex is a vertex in the first half of the path (not including the midpoint) that is a mirror image of a vertex in the second half, with respect to reflection along the vertical line through the midpoint of the path.

			For n=5 there are 4 Dyck paths with 2 symmetric vertices: uuuuddddud, uduuuudddd, uuudddudud, ududuuuddd.
Triangle begins:
  1;
  0,   2;
  0,   2,    3;
  0,   2,    6,    6;
  0,   4,   12,   16,   10;
  0,   8,   24,   40,   40,   20;
  0,  20,   60,  104,  120,   90,   35;
  0,  50,  150,  270,  350,  330,  210,   70;
  0, 140,  420,  768, 1040, 1080,  840,  448,  126;
  0, 392, 1176, 2184, 3080, 3468, 3108, 2128, 1008, 252;
  ...

Alois P. Heinz, Rows n = 1..200, flattened
Sergi Elizalde, The degree of symmetry of lattice paths, arXiv:2002.12874 [math.CO], 2020.
Sergi Elizalde, Measuring symmetry in lattice paths and partitions, Sem. Lothar. Combin. 84B.26, 12 pp (2020).

Row sums give A000108.
Main diagonal gives A001405.
Column k=2 gives 2*A005817(n-3) (for n>2).

b:= proc(x, y, v) option remember; expand(
      `if`(min(y, v, x-max(y, v))<0, 0, `if`(x=0, 1, (l-> add(add(
      `if`(y+i=v+j, z, 1)*b(x-1, y+i, v+j), i=l), j=l))([-1, 1]))))
    end:
g:= proc(n) option remember; add(b(n, j$2), j=0..n) end:
T:= (n, k)-> coeff(g(n), z, k):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Feb 12 2021

b[x_, y_, v_] := b[x, y, v] = Expand[If[Min[y, v, x - Max[y, v]] < 0, 0, If[x == 0, 1, Function[l, Sum[Sum[If[y + i == v + j, z, 1]*b[x - 1, y + i, v + j], {i, l}], {j, l}]][{-1, 1}]]]];
g[n_] := g[n] = Sum[b[n, j, j], {j, 0, n}];
T[n_, k_] := Coefficient[g[n], z, k];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Feb 13 2021, after Alois P. Heinz *)

A341415 Triangle read by rows: T(n,k) is the number of grand Dyck paths of semilength n having degree of symmetry k (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 0, 2, 2, 0, 4, 4, 8, 0, 8, 14, 16, 24, 0, 16, 44, 64, 48, 64, 0, 32, 148, 208, 216, 128, 160, 0, 64, 504, 736, 720, 640, 320, 384, 0, 128, 1750, 2592, 2672, 2176, 1760, 768, 896, 0, 256, 6156, 9280, 9696, 8448, 6080, 4608, 1792, 2048, 0, 512

Offset: 0

Sergi Elizalde, Feb 12 2021

The degree of symmetry of a grand Dyck path is defined as the number of steps in the first half that are mirror images of steps in the second half, with respect to the reflection along a vertical line through the midpoint of the path.

			For n=3 there are 4 grand Dyck paths with degree of symmetry equal to 0, namely uddduu, uudddu, duuudd, dduuud.
The triangle begins:
    1
    0    2
    2    0    4
    4    8    0    8
   14   16   24    0   16
   44   64   48   64    0   32
  148  208  216  128  160    0  64
  504  736  720  640  320  384   0  128

Sergi Elizalde, The degree of symmetry of lattice paths, arXiv:2002.12874 [math.CO], 2020.
Sergi Elizalde, Measuring symmetry in lattice paths and partitions, Sem. Lothar. Combin. 84B.26, 12 pp (2020).

Cf. A000079 (diagonal), A000984 (row sums).

G.f.: 1/(2(1-u)z+sqrt(1-4z)).

A275448 The number of weakly alternating bargraphs of semiperimeter n. A bargraph is said to be weakly alternating if its ascents and descents alternate. An ascent (descent) is a maximal sequence of consecutive U (D) steps.

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 28, 65, 146, 327, 749, 1756, 4165, 9913, 23652, 56687, 136627, 330969, 804915, 1963830, 4805523, 11793046, 29019930, 71589861, 177006752, 438561959, 1088714711, 2707615555, 6745272783, 16830750107, 42058592797, 105248042792

Offset: 2

Emeric Deutsch, Sergi Elizalde, Aug 26 2016

nonn

			a(4)=3 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 only [1,1,1],[2,2], and [3] lead to weakly alternating bargraphs.

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. A082582, A023432.

g := ((1-3*z+3*z^2-sqrt((1-3*z+z^2)*(1-3*z+5*z^2-4*z^3)))*(1/2))/(z*(1-z)): gser:= series(g,z=0,43): seq(coeff(gser,z,n), n=2..40);

terms = 32;
g[z_] = ((1 - 3z + 3z^2 - Sqrt[(1 - 3z + z^2)(1 - 3z + 5z^2 - 4z^3)])*(1/2) )/(z(1-z));
Drop[CoefficientList[g[z] + O[z]^(terms+2), z], 2] (* Jean-François Alcover, Aug 07 2018 *)

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

D-finite with recurrence (n+1)*a(n) +(-7*n+2)*a(n-1) +3*(7*n-11)*a(n-2) +(-37*n+107)*a(n-3) +3*(13*n-54)*a(n-4) +3*(-7*n+37)*a(n-5) +2*(2*n-13)*a(n-6)=0. - R. J. Mathar, Jul 22 2022

A274495 The length of the longest initial sequence of the form UHUH..., summed over all bargraphs having semiperimeter n (n>=2).

Original entry on oeis.org

2, 3, 9, 23, 62, 171, 482, 1384, 4036, 11924, 35619, 107407, 326521, 999675, 3079634, 9539366, 29693294, 92831327, 291366477, 917765199, 2900217452, 9192097510, 29213057684, 93073003438, 297215560553, 951144390092, 3049877146281, 9797605279905

Offset: 2

Emeric Deutsch, Sergi Elizalde, Aug 26 2016

nonn

			a(4) = 9 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 the sum of the lengths of their longest initial sequence of the form UHUH... is 2+4+1+1+1.

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. A082582, A274494.

Q := sqrt((1-z)*(1-3*z-z^2-z^3)): g := (((1-z)*(1-4*z^2-3*z^3-2*z^4)-(1+z-z^2-2*z^3)*Q)*(1/2))/(z*(1-z)): gser := series(g, z = 0, 38): seq(coeff(gser, z, n), n = 2 .. 34);

terms = 28;
g[z_] = (((1-z)(1 - 4z^2 - 3z^3 - 2z^4) - (1 + z - z^2 - 2z^3)*Q)(1/2))/(z (1-z)) /. Q -> Sqrt[(1-z)(1 - 3z - z^2 - z^3)];
Drop[CoefficientList[g[z] + O[z]^(terms+2), z], 2] (* Jean-François Alcover, Aug 07 2018 *)

G.f.: g(z) = ((1-z)*(1-4*z^2-3*z^3-2*z^4)-(1+z-z^2-2*z^3)*Q)/(2*z*(1-z)), where Q = sqrt((1-z)*(1-3*z-z^2-z^3)).
a(n) = Sum_{k>=1} k*A274494(n,k).
D-finite with recurrence -(n+1)*(19*n-44)*a(n) +n*(43*n-65)*a(n-1) +2*(47*n^2-289*n+342)*a(n-2) +2*(-33*n^2+170*n-61)*a(n-3) +(-19*n^2+87*n+22)*a(n-4) -(33*n-31)*(n-8)*a(n-5)=0. - R. J. Mathar, Jul 22 2022

A274494 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k as the length of the longest initial sequence of the form UHUH... (n>=2, 1<=k<=2*floor(n/2)).

Original entry on oeis.org

0, 1, 1, 1, 3, 1, 0, 1, 8, 2, 1, 2, 22, 5, 4, 3, 0, 1, 62, 13, 12, 6, 1, 3, 178, 35, 35, 15, 5, 6, 0, 1, 519, 97, 103, 40, 17, 13, 1, 4, 1533, 275, 306, 110, 53, 33, 6, 10, 0, 1, 4578, 794, 917, 310, 163, 90, 23, 24, 1, 5, 13800, 2327, 2770, 891, 501, 253, 77, 63, 7, 15, 0, 1

Offset: 2

Emeric Deutsch, Sergi Elizalde, Aug 26 2016

Number of entries in row n is 2*floor(n/2).
Sum of entries in row n = A082582(n).
Sum(k*T(n,k),k>=0) = A274495(n).

			Row 4 is 3,1,0,1 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 the lengths of the longest initial sequence of the form UHUH... are 2,4,1,1,1, respectively.
Triangle starts
0,1;
1,1;
3,1,0,1;
8,2,1,2;
22,5,4,3,0,1;

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 [math.CO], 2016.

Cf. A082582, A274495.

a := z*(1-t^2*z-t^2*z^3+t^4*z^3): b := -t*(1-3*z+z^2+t*z^2-t^2*z^2-z^3+2*t^2*z^3+t*z^4-2*t^3*z^4+t^2*z^4): c := t^2*z^2*(t+z-2*t*z-t*z^2+t^2*z^2): eq := a*G^2+b*G+c = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 21)): for n from 2 to 18 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 2 to 18 do seq(coeff(P[n], t, j), j = 1 .. 2*floor((1/2)*n)) end do; # yields sequence in triangular form

nmax = 12;
a = z (1 - t^2 z - t^2 z^3 + t^4 z^3);
b = -t (1 - 3z + z^2 + t z^2 - t^2 z^2 - z^3 + 2t^2 z^3 + t z^4 - 2t^3 z^4 + t^2 z^4);
c = t^2 z^2 (t + z - 2t z - t z^2 + t^2 z^2);
G = 0; Do[G = Series[(-c - a G^2)/b, {z, 0, nmax}, {t, 0, nmax}] // Normal, {nmax}];
cc = CoefficientList[G, z];
row[n_] := CoefficientList[cc[[n+1]], t] // Rest;
Table[row[n], {n, 2, nmax}] // Flatten (* Jean-François Alcover, Jul 25 2018 *)

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

A276066 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having a total of k double rises and double falls (n>=2,k>=0). A double rise (fall) in a bargraph is any pair of adjacent up (down) steps.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 0, 1, 2, 4, 1, 4, 1, 0, 1, 4, 6, 8, 8, 1, 6, 1, 0, 1, 7, 14, 22, 12, 19, 12, 1, 8, 1, 0, 1, 13, 34, 43, 48, 55, 18, 35, 16, 1, 10, 1, 0, 1, 26, 72, 105, 148, 109, 116, 103, 24, 56, 20, 1, 12, 1, 0, 1, 52, 154, 276, 344, 347, 398, 205, 232, 166, 30, 82, 24, 1, 14, 1, 0, 1

Offset: 2

Emeric Deutsch and Sergi Elizalde, Aug 25 2016

Number of entries in row n is 2n-3.
Sum of entries in row n = A082582(n).
T(n,0) = A023431(n-2) = A025246(n+1).
Sum(k*T(n,k),k>=0) = 2*A273714(n).

			Row 4 is 1,2,1,0,1 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 a total of  0, 1, 1, 2, 4 double rises and double falls, respectively.
Triangle starts:
1;
1,0,1;
1,2,1,0,1;
2,4,1,4,1,0,1;
4,6,8,8,1,6,1,0,1.

Alois P. Heinz, Rows n = 2..140, 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. A023431, A025246, A082582, A273714.

eq := z*G^2-(1-z-t^2*z-2*t*z^2+t^2*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 .. 2*n-4) end do; # yields sequence in triangular form.
# second Maple program:
b:= proc(n, y, t) option remember; expand(`if`(n=0, (1-t)*
      z^(y-1), `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`(t=-1, z, 1))+
     `if`(y<1, 0, b(n-1, y, 0))))
    end:
T:= n->(p->seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=2..12);  # Alois P. Heinz, Aug 25 2016

b[n_, y_, t_] := b[n, y, t] = Expand[If[n == 0, (1 - t)*z^(y - 1), 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[t == -1, z, 1]] + If[y < 1, 0, b[n - 1, y, 0]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, 0]]; Table[T[n], {n, 2, 12}] // Flatten (* Jean-François Alcover, Dec 02 2016 after Alois P. Heinz *)

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

The g.f. B(t,s,z) of bargraphs, where t(s) marks double rises (falls) and z marks semiperimeter, satisfies zB^2 - (1-(1+ts)z +(ts- t-s)z^2)B + z^2 = 0.

A274493 Number of bargraphs of semiperimeter n having no horizontal segments of length 1 (n>=2). By a horizontal segment of length 1 we mean a horizontal step that is not adjacent to any other horizontal step.

Original entry on oeis.org

0, 1, 2, 3, 6, 13, 27, 57, 123, 267, 584, 1289, 2864, 6399, 14373, 32435, 73498, 167175, 381551, 873541, 2005622, 4616895, 10653607, 24638263, 57097885, 132575577, 308378460, 718506295, 1676706422, 3918515001, 9170350093, 21488961641, 50417138776, 118425429213, 278476687643

Offset: 2

Emeric Deutsch and Sergi Elizalde, Jun 27 2016

			a(4)=2 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 pictures give the values 0,2,2,0,1 for the number of horizontal segments of length 1.

M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Cf. A082582, A274491.

g:=((1-2*z+z^2-2*z^3-sqrt((1-z)*(1-3*z+3*z^2-5*z^3+4*z^4-4*z^5)))*(1/2))/z^2: gser:=series(g,z=0,40): seq(coeff(gser,z,n),n=2..36);

a(n) = A274491(n,0).
G.f.: g(z)=(1-2z+z^2-2z^3-sqrt((1-z)(1-3z+3z^2-5z^3+4z^4-4z^5)))/(2z^2).
D-finite with recurrence (n+2)*a(n) +2*(-2*n-1)*a(n-1) +6*(n-1)*a(n-2) +4*(-2*n+5)*a(n-3) +9*(n-4)*a(n-4) +4*(-2*n+11)*a(n-5) +4*(n-7)*a(n-6)=0. - R. J. Mathar, Jul 22 2022

A274492 Number of horizontal segments of length 1 in all bargraphs of semiperimeter n (n>=2). By a horizontal segment of length 1 we mean a horizontal step that is not adjacent to any other horizontal step.

Original entry on oeis.org

1, 1, 5, 16, 52, 170, 556, 1821, 5973, 19620, 64536, 212553, 700903, 2313864, 7646670, 25294673, 83748689, 277518319, 920332567, 3054319120, 10143305864, 33707066667, 112078220233, 372875904038, 1241182355688, 4133534991928, 13772413826888, 45908128269573

Offset: 2

Emeric Deutsch and Sergi Elizalde, Jun 27 2016

nonn

			a(4)=5 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 pictures give the values 0,2,2,0,1 for the number of horizontal segments of length 1.

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. A082582, A274491.

g:=(1/2)*(1-z)^3*(1-2*z-z^2-Q)/(z*Q): Q:=sqrt((1-z)*(1-3*z-z^2-z^3)): gser:= series(g,z=0,30): seq(coeff(gser,z,n), n=2..27);

g = (1-z)^3 (1-2z-z^2-Q)/(2z Q) /. Q -> Sqrt[(1-z)(1-3z-z^2-z^3)];
a[n_] := SeriesCoefficient[g, {z, 0, n}];
Table[a[n], {n, 2, 29}] (* Jean-François Alcover, Jul 25 2018 *)

G.f.: g(z)=(1-z)^3*(1-2z-z^2-Q)/(2zQ), where Q = sqrt((1-z)(1-3z-z^2-z^3)).
a(n) = Sum(k*A274491(n,k), k>=0).
D-finite with recurrence (n+1)*a(n) +3*(-2*n+1)*a(n-1) +3*(3*n-8)*a(n-2) +6*(1)*a(n-3) +(-n+19)*a(n-4) +(-2*n+15)*a(n-5) +(-n+8)*a(n-6)=0. - R. J. Mathar, Jul 22 2022

cp's OEIS Frontend

User: Sergi Elizalde

A376110 Number of permutations of {1,2,...,n} that are non-self-overlapping as Hertzsprung patterns; also called non-extendible.

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A341445 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having degree of symmetry k (n >= 1, 1 <= k <= n).

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A339754 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k symmetric vertices (n >= 1, 1 <= k <= n).

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A341415 Triangle read by rows: T(n,k) is the number of grand Dyck paths of semilength n having degree of symmetry k (n >= 0, 0 <= k <= n).

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A275448 The number of weakly alternating bargraphs of semiperimeter n. A bargraph is said to be weakly alternating if its ascents and descents alternate. An ascent (descent) is a maximal sequence of consecutive U (D) steps.

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A274495 The length of the longest initial sequence of the form UHUH..., summed over all bargraphs having semiperimeter n (n>=2).

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A274494 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k as the length of the longest initial sequence of the form UHUH... (n>=2, 1<=k<=2*floor(n/2)).

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A276066 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having a total of k double rises and double falls (n>=2,k>=0). A double rise (fall) in a bargraph is any pair of adjacent up (down) steps.

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