A291883 -id:A291883 - cp's OEIS frontend

A056326 Number of reversible string structures with n beads using exactly two different colors.

0, 1, 2, 5, 9, 19, 35, 71, 135, 271, 527, 1055, 2079, 4159, 8255, 16511, 32895, 65791, 131327, 262655, 524799, 1049599, 2098175, 4196351, 8390655, 16781311, 33558527, 67117055, 134225919, 268451839, 536887295, 1073774591, 2147516415, 4295032831, 8590000127

Offset: 1

Marks R. Nester

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3, 0, -6, 4).

Column k=2 of A284949 and of A291883.

Cf. A056309.

Table[(StirlingS2[n,2] + StirlingS2[Floor[n/2]+1,2])/2, {n,1,30}] (* Robert A. Russell, Jan 29 2018 *)
LinearRecurrence[{3, 0, -6, 4}, {0, 1, 2, 5}, 35] (* or *)
Rest@ CoefficientList[Series[x^2*(x^2 + x - 1)/((x - 1) (2 x - 1) (2 x^2 - 1)), {x, 0, 35}], x] (* Michael De Vlieger, Jan 31 2018 *)

a(n) = A005418(n) - 1.
From Colin Barker, Nov 25 2012: (Start)
a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4).
G.f.: x^2*(x^2+x-1)/((x-1)*(2*x-1)*(2*x^2-1)). (End)

A291885 Number of symmetrically unique Dyck paths of semilength 2n and height n.

Original entry on oeis.org

1, 1, 5, 31, 252, 2117, 18546, 164229, 1469596, 13229876, 119712521, 1087573357, 9914033252, 90633332870, 830621140260, 7628813061585, 70200092854044, 647070588612140, 5973385906039684, 55217660246861884, 511054426374819184, 4735208302827742549

Offset: 0

Alois P. Heinz, Sep 05 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1026

Cf. A291883.

b:= proc(x, y, k) option remember; `if`(x=0, 1, `if`(y+1<=min(k,
        x-1), b(x-1, y+1, k), 0)+`if`(y>0, b(x-1, y-1, k), 0))
    end:
g:= proc(x, y, k) option remember; `if`(x=0, 1, `if`(y>0,
      g(x-2, y-1, k), 0)+ `if`(y+1<=k, g(x-2, y+1, k), 0))
    end:
a:= n-> `if`(n=0, 1, (b(4*n, 0, n)  +g(4*n, 0, n)
                     -b(4*n, 0, n-1)-g(4*n, 0, n-1))/2):
seq(a(n), n=0..30);

b[x_, y_, k_] := b[x, y, k] = If[x == 0, 1, If[y + 1 <= Min[k, x - 1], b[x - 1, y + 1, k], 0] + If[y > 0, b[x - 1, y - 1, k], 0]];
g[x_, y_, k_] := g[x, y, k] = If[x == 0, 1, If[y > 0, g[x - 2, y - 1, k], 0] + If[y + 1 <= k, g[x - 2, y + 1, k], 0]];
a[n_] := If[n == 0, 1, (b[4n, 0, n] + g[4n, 0, n] - b[4n, 0, n - 1] - g[4n, 0, n - 1])/2];
Array[a, 30, 0] (* Jean-François Alcover, May 31 2019, after Alois P. Heinz *)

from sympy.core.cache import cacheit
@cacheit
def b(x, y, k): return 1 if x==0 else (b(x - 1, y + 1, k) if y + 1<=min(k, x - 1) else 0) + (b(x - 1, y - 1, k) if y>0 else 0)
@cacheit
def g(x, y, k): return 1 if x==0 else (g(x - 2, y - 1, k) if y>0 else 0) + (g(x - 2, y + 1, k) if y + 1<=k else 0)
def a(n): return 1 if n==0 else (b(4*n, 0, n) + g(4*n, 0, n) - b(4*n, 0, n - 1) - g(4*n, 0, n - 1))//2
print([a(n) for n in range(31)]) # Indranil Ghosh, Sep 06 2017

a(n) = A291883(2n,n).

A291886 Total height of all symmetrically unique Dyck paths of semilength n.

Original entry on oeis.org

0, 1, 3, 8, 24, 73, 239, 808, 2839, 10173, 37107, 136778, 508622, 1902931, 7156180, 27020991, 102387426, 389129477, 1482851762, 5664065704, 21681357734, 83154466615, 319486977769, 1229491584784, 4738571829454, 18288152053984, 70672285064968, 273430196389027

Offset: 0

Alois P. Heinz, Sep 05 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..600

Cf. A291883.

a(n) = Sum_{k=1..n} k * A291883(n,k)

A291887 Number of symmetrically unique Dyck paths of semilength n and height three.

Original entry on oeis.org

1, 3, 11, 31, 91, 250, 690, 1863, 5017, 13389, 35621, 94354, 249352, 657421, 1730685, 4550031, 11950963, 31365810, 82274126, 215711947, 565376593, 1481448697, 3881041065, 10165822434, 26624745964, 69725071281, 182583838181, 478093234315, 1251829592839

Offset: 3

Alois P. Heinz, Sep 05 2017

Alois P. Heinz, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (6, -9, -8, 27, -10, -10, 4).

Column k=3 of A291883.

G.f.: -x^3*(x^4+2*x^2-3*x+1)/((2*x-1)*(x^2-3*x+1)*(2*x^2-1)*(x^2+x-1)).

A291888 Number of symmetrically unique Dyck paths of semilength n and height four.

Original entry on oeis.org

1, 4, 19, 69, 252, 855, 2867, 9339, 30085, 95524, 300796, 939228, 2915481, 9001380, 27675779, 84782673, 258956584, 788926699, 2398385803, 7277878707, 22050326805, 66718218120, 201640080056, 608808949752, 1836608777233, 5536482441124, 16679237438035

Offset: 4

Alois P. Heinz, Sep 05 2017

Alois P. Heinz, Table of n, a(n) for n = 4..1000
Index entries for linear recurrences with constant coefficients, signature (8,-19,-2,66,-76,3,30,-9).

Column k=4 of A291883.

G.f.: x^4*(3*x^4-5*x^3+6*x^2-4*x+1)/((x-1)*(3*x-1)*(x^2-3*x+1)*(3*x^2-1)*(x^2+x-1)).

A291889 Number of symmetrically unique Dyck paths of semilength n and height five.

Original entry on oeis.org

1, 5, 29, 127, 540, 2117, 8063, 29688, 107237, 380549, 1333709, 4624308, 15899625, 54278502, 184209808, 622029353, 2091461574, 7006165827, 23394683289, 77899483723, 258750654760, 857596024194, 2836911700038, 9368339478043, 30889646001696, 101710568454396

Offset: 5

Alois P. Heinz, Sep 05 2017

Alois P. Heinz, Table of n, a(n) for n = 5..1000
Index entries for linear recurrences with constant coefficients, signature (10, -33, 20, 113, -234, 66, 224, -240, 84, -9).

Column k=5 of A291883.

G.f.: -x^5*(18*x^3-14*x^4+3*x^5-12*x^2+5*x-1) / ((x-1) *(3*x-1) *(3*x^2-1) *(x^3-2*x^2-x+1) *(x^3-6*x^2+5*x-1)).

A291890 Number of symmetrically unique Dyck paths of semilength n and height six.

Original entry on oeis.org

1, 6, 41, 209, 1005, 4411, 18546, 74854, 294293, 1131635, 4280686, 15974526, 58974616, 215772740, 783598381, 2827768273, 10149744903, 36261522803, 129027358151, 457486923079, 1617031493924, 5699720240694, 20040736126691, 70308818376855, 246171461366950

Offset: 6

Alois P. Heinz, Sep 05 2017

Alois P. Heinz, Table of n, a(n) for n = 6..1000
Index entries for linear recurrences with constant coefficients, signature (12, -51, 66, 144, -540, 411, 510, -1006, 380, 282, -284, 84, -8).

Column k=6 of A291883.

G.f.: x^6*(20*x^2-43*x^3+48*x^4-20*x^5+x^7-6*x+1) / ((2*x-1) *(2*x^2-4*x+1) *(x^3-2*x^2-x+1) *(x^3-6*x^2+5*x-1) *(2*x^4-4*x^2+1)).

A291891 Number of symmetrically unique Dyck paths of semilength n and height seven.

Original entry on oeis.org

1, 7, 55, 319, 1705, 8238, 37674, 164229, 692627, 2843282, 11433826, 45212792, 176385132, 680452948, 2600725892, 9862321095, 37150333241, 139139984973, 518538211261, 1924077739700, 7112221384554, 26201080984497, 96233327019085, 352501632479306

Offset: 7

Alois P. Heinz, Sep 05 2017

Alois P. Heinz, Table of n, a(n) for n = 7..1000
Index entries for linear recurrences with constant coefficients, signature (13, -60, 84, 203, -807, 569, 1173, -2090, 480, 1046, -662, -16, 76, -8).

Column k=7 of A291883.

CoefficientList[Series[-(6*x - 24*x^2 + 60*x^3 - 67*x^4 + 21*x^5 + x^6 + x^7 - 1)/((x - 1)*(2*x - 1)*(2*x^2 - 4*x + 1)*(x^3 + 3*x^2 - 1)*(x^3 - 9*x^2 + 6*x - 1)*(2*x^4 - 4*x^2 + 1)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Oct 14 2023 *)

G.f.: -x^7*(6*x-24*x^2+60*x^3-67*x^4+21*x^5+x^6+x^7-1) / ((x-1) *(2*x-1) *(2*x^2-4*x+1) *(x^3+3*x^2-1) *(x^3-9*x^2+6*x-1) *(2*x^4-4*x^2+1)).

A291892 Number of symmetrically unique Dyck paths of semilength n and height eight.

Original entry on oeis.org

1, 8, 71, 461, 2706, 14235, 70161, 327469, 1469596, 6389144, 27097948, 112630404, 460511702, 1857372265, 7406424903, 29250500171, 114576069911, 445647539540, 1722814022086, 6624828067482, 25356181172529, 96650473757117, 367059233827762, 1389476976680608

Offset: 8

Alois P. Heinz, Sep 05 2017

Alois P. Heinz, Table of n, a(n) for n = 8..1000
Index entries for linear recurrences with constant coefficients, signature (15, -84, 178, 160, -1436, 1985, 1447, -6220, 4096, 3445, -5665, 1925, 405, -275, 25).

Column k=8 of A291883.

G.f.: x^8*(-1+7*x-35*x^2+110*x^3-171*x^4+113*x^5-25*x^6+5*x^7) / ((x-1) *(x^2-3*x+1) *(5*x^2-5*x+1) *(x^3+3*x^2-1) *(x^3-9*x^2+6*x-1) *(5*x^4-5*x^2+1)).

A291893 Number of symmetrically unique Dyck paths of semilength n and height nine.

Original entry on oeis.org

1, 9, 89, 639, 4082, 23191, 122381, 607944, 2887756, 13229876, 58898404, 256121253, 1092452270, 4585149401, 18985099139, 77707979700, 314944855135, 1265638606115, 5048705160315, 20010229736469, 78861912700117, 309252947330313, 1207359268831473

Offset: 9

Alois P. Heinz, Sep 05 2017

Alois P. Heinz, Table of n, a(n) for n = 9..1000
Index entries for linear recurrences with constant coefficients, signature (18, -129, 428, -339, -2154, 7022, -5004, -13608, 31128, -13587, -27364, 38529, -11230, -11610, 11180, -3805, 550, -25).

Column k=9 of A291883.

G.f.: -x^9*(-1+9*x-56*x^2+230*x^3-548*x^4+741*x^5-564*x^6+238*x^7-55*x^8+5*x^9) / ((x^2-3*x+1) *(5*x^2-5*x+1) *(5*x^4-5*x^2+1) *(x^5-3*x^4-3*x^3+4*x^2+x-1) *(x^5-15*x^4+35*x^3-28*x^2+9*x-1)).

cp's OEIS Frontend

A056326 Number of reversible string structures with n beads using exactly two different colors.

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A291885 Number of symmetrically unique Dyck paths of semilength 2n and height n.

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A291886 Total height of all symmetrically unique Dyck paths of semilength n.

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A291887 Number of symmetrically unique Dyck paths of semilength n and height three.

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A291888 Number of symmetrically unique Dyck paths of semilength n and height four.

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A291889 Number of symmetrically unique Dyck paths of semilength n and height five.

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A291890 Number of symmetrically unique Dyck paths of semilength n and height six.

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A291891 Number of symmetrically unique Dyck paths of semilength n and height seven.

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A291892 Number of symmetrically unique Dyck paths of semilength n and height eight.

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A291893 Number of symmetrically unique Dyck paths of semilength n and height nine.

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