A057585 -id:A057585 - cp's OEIS frontend

A333498 Sum of the heights of all Motzkin paths of length n.

0, 0, 1, 3, 9, 25, 70, 196, 552, 1560, 4423, 12573, 35826, 102310, 292786, 839554, 2411945, 6941593, 20011328, 57779038, 167069317, 483739961, 1402413161, 4070537585, 11827842021, 34403798725, 100167396088, 291903951462, 851380987390, 2485175809878

Offset: 0

Alois P. Heinz, Mar 24 2020

nonn

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

Cf. A001006, A057585, A097862, A136439, A333504, A333608.

b:= proc(x, y, h) option remember; `if`(x=0, h, add(
      b(x-1, y+j, max(h, y)), j=-min(1, y)..min(1, x-y-1)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..35);

b[x_, y_, h_] := b[x, y, h] = If[x == 0, h, Sum[b[x - 1, y + j, Max[h, y]], {j, -Min[1, y], Min[1, x - y - 1]}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 35] (* Jean-François Alcover, May 10 2020, after Maple *)

a(n) = Sum_{k=1..floor(n/2)} k * A097862(n,k).

A129181 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n such that the area between the x-axis and the path is k (n>=0; 0<=k<=floor(n^2/4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 1, 4, 6, 4, 3, 2, 1, 1, 5, 10, 10, 8, 7, 5, 3, 1, 1, 1, 6, 15, 20, 19, 18, 16, 12, 8, 6, 3, 2, 1, 1, 7, 21, 35, 40, 41, 41, 36, 29, 23, 18, 12, 9, 5, 3, 1, 1, 1, 8, 28, 56, 76, 86, 93, 92, 83, 72, 62, 50, 40, 30, 22, 14, 10, 6, 3, 2, 1, 1, 9, 36, 84, 133, 168

Offset: 0

Emeric Deutsch, Apr 08 2007

Row n has 1+floor(n^2/4) terms.

Row sums are the Motzkin numbers (A001006).

			T(5,3) = 4 because we have LULLD, ULLDL, UDULD and ULDUD, where U=(1,1), L=(1,0) and D=(1,-1).
Triangle starts:
00: 1;
01: 1;
02: 1, 1;
03: 1, 2,  1;
04: 1, 3,  3,  1,  1;
05: 1, 4,  6,  4,  3,  2,  1;
06: 1, 5, 10, 10,  8,  7,  5,  3, 1, 1;
07: 1, 6, 15, 20, 19, 18, 16, 12, 8, 6, 3, 2, 1;
...
From _Joerg Arndt_, Apr 19 2014: (Start)
Row n=5 corresponds to the following Motzkin paths (dots denote zeros):
# :   height in path   area    step in path
01:  [ . . . . . . ]     0     0 0 0 0 0
02:  [ . . . . 1 . ]     1     0 0 0 + -
03:  [ . . . 1 . . ]     1     0 0 + - 0
04:  [ . . . 1 1 . ]     2     0 0 + 0 -
05:  [ . . 1 . . . ]     1     0 + - 0 0
06:  [ . . 1 . 1 . ]     2     0 + - + -
07:  [ . . 1 1 . . ]     2     0 + 0 - 0
08:  [ . . 1 1 1 . ]     3     0 + 0 0 -
09:  [ . . 1 2 1 . ]     4     0 + + - -
10:  [ . 1 . . . . ]     1     + - 0 0 0
11:  [ . 1 . . 1 . ]     2     + - 0 + -
12:  [ . 1 . 1 . . ]     2     + - + - 0
13:  [ . 1 . 1 1 . ]     3     + - + 0 -
14:  [ . 1 1 . . . ]     2     + 0 - 0 0
15:  [ . 1 1 . 1 . ]     3     + 0 - + -
16:  [ . 1 1 1 . . ]     3     + 0 0 - 0
17:  [ . 1 1 1 1 . ]     4     + 0 0 0 -
18:  [ . 1 1 2 1 . ]     5     + 0 + - -
19:  [ . 1 2 1 . . ]     4     + + - - 0
20:  [ . 1 2 1 1 . ]     5     + + - 0 -
21:  [ . 1 2 2 1 . ]     6     + + 0 - -
(End)

Alois P. Heinz, Rows n = 0..50, flattened
Clementa Alonso-González and Miguel Ángel Navarro-Pérez, Motzkin numbers and flag codes, arXiv:2207.01997 [math.CO], 2022.
Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, Motzkin and Catalan Tunnel Polynomials, J. Int. Seq., Vol. 21 (2018), Article 18.8.8.
Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, Consecutive patterns in restricted permutations and involutions, arXiv:1902.02213 [math.CO], 2019.
A. Bärtschi, B. Geissmann, D. Graf, T. Hruz, P. Penna, and T. Tschager, On computing the total displacement number via weighted Motzkin paths, arXiv:1606.05538 [cs.DS], 2016.
Thomas Grubb and Frederick Rajasekaran, Set Partition Patterns and the Dimension Index, arXiv:2009.00650 [math.CO], 2020. Mentions this sequence.

Cf. A001006, A057585.

Antidiagonal sums give A186085(n+1).

G:=1/(1-z-t*z^2*g[1]): for i from 1 to 13 do g[i]:=1/(1-t^i*z-t^(2*i+1)*z^2*g[i+1]) od: g[14]:=0: Gser:=simplify(series(G,z=0,13)): for n from 0 to 10 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 10 do seq(coeff(P[n],t,j),j=0..floor(n^2/4)) od; # yields sequence in triangular form
# second Maple program
b:= proc(x, y, k) option remember;
      `if`(x<0 or xx^2, 0,
      `if`(x=0, 1, add(b(x-1, y+i, k-y-i/2), i=-1..1)))
    end:
T:= (n, k)-> b(n, 0, k):
seq(seq(T(n, k), k=0..floor(n^2/4)), n=0..12); # Alois P. Heinz, Jun 28 2012

b[x_, y_, k_] := b[x, y, k] = If[x<0 || xx^2, 0, If[x==0, 1, Sum[b[x-1, y+i, k-y-i/2], {i, -1, 1}]]]; T[n_, k_] := b[n, 0, k]; Table[Table[ T[n, k], {k, 0, Floor[n^2/4]}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *)

G.f. G(t,z) satisfies G(t,z) = 1 + z*G(t,z) + t*z^2*G(t,t*z)*G(t,z).
Sum_{k>=0} k * T(n,k) = A057585(n).
Sum_{j=0..n} T(n-j,j) = A186085(n+1). - Alois P. Heinz, Jun 25 2023

A367778 a(n) is the sum of the squares of the areas under Motzkin paths of length n.

Original entry on oeis.org

0, 1, 6, 40, 198, 910, 3848, 15492, 59920, 224917, 824074, 2960828, 10466610, 36498195, 125801144, 429284612, 1452174984, 4874940295, 16254780970, 53873727516, 177594715034, 582603630260, 1902860189328, 6190199896600, 20064013907288, 64815504118695, 208739559416878, 670345766842528

Offset: 1

AJ Bu, Nov 29 2023

nonn

a(n) is the sum of the squares of the areas under Motzkin paths of length n (nonnegative walks beginning and ending in 0, with jumps -1,0,+1).

			a(3) = 6 = 1*2^2 + 2*1^2 because there is 1 Motzkin path of length 3 with area 2 and 2 Motzkin paths of length 3 with area 1.

AJ Bu, Explicit Generating Functions for the Sum of the Areas Under Dyck and Motzkin Paths (and for Their Powers), arXiv:2310.17026 [math.CO], 2023.

Cf. A001006, A057585.

G:=((x - 1 + sqrt(-(x + 1)*(3*x - 1)))*(3*sqrt(-(x + 1)*(3*x - 1))*x^4 - 9*x^5 - 14*sqrt(-(x + 1)*(3*x - 1))*x^3 + 15*x^4 + 8*sqrt(-(x + 1)*(3*x - 1))*x^2 + 26*x^3 + 4*sqrt(-(x + 1)*(3*x - 1))*x - 4*x^2 - sqrt(-(x + 1)*(3*x - 1)) - 5*x + 1))/( 4*(x + 1)^3*(3*x - 1)^3*x^2):  Gser:=series(G, x=0, 30): seq(coeff(Gser,x,n), n=1..26);

seq(n) = {my(w=sqrt((1 + x)*(1 - 3*x) + O(x*x^n))); Vec((1 - x - w)*(w^2*(1 - 3*x - 7*x^2 + 3*x^3) - w*(1 - x)*(1 - 3*x - 11*x^2 + 3*x^3))/(2*w^3*x)^2, -n)} \\ Andrew Howroyd, Jan 07 2024

G.f.: (1 - x - w)*(w^2*(1 - 3*x - 7*x^2 + 3*x^3) - w*(1 - x)*(1 - 3*x - 11*x^2 + 3*x^3))/(2*w^3*x)^2 where w is sqrt((1 + x)*(1 - 3*x)).

D-finite with recurrence -(n+2)*(37012171*n -222599339)*a(n) +3*(n+1)*(108071243*n -631482704)*a(n-1) +(-512534971*n^2 +2421530181*n +1780794712)*a(n-2) +3*(-641100693*n^2 +4745437175*n -5322233482)*a(n-3) +(4162359143*n^2 -33175360881*n +59296953526)*a(n-4) +3*(1437180249*n^2 -9681487559*n +8357806732)*a(n-5) +9*(-754462425*n^2 +6932112703*n -14939114852)*a(n-6) -27*(218140823*n -693079002)*(n-5)*a(n-7)=0. - R. J. Mathar, Jan 11 2024

A367779 a(n) is the sum of the cubed areas under Motzkin paths of length n.

Original entry on oeis.org

0, 0, 1, 10, 118, 818, 5092, 27564, 137836, 644836, 2870189, 12266918, 50724954, 204046142, 801892081, 3089123960, 11696423536, 43623049688, 160547844283, 583940294930, 2101624362838, 7492542382034, 26484322064854, 92891831844644, 323514376584988, 1119432296516028, 3850521166068067

Offset: 0

AJ Bu, Nov 29 2023

nonn

a(n) is the sum of the cubed areas under Motzkin paths of length n (nonnegative walks beginning and ending in 0, with jumps -1, 0, +1).

AJ Bu, Explicit Generating Functions for the Sum of the Areas Under Dyck and Motzkin Paths (and for Their Powers), arXiv:2310.17026 [math.CO], 2023.

Cf. A001006, A057585.

G:= ((x - 1 + sqrt(-3*x^2 - 2*x + 1))*(27*sqrt(-3*x^2 - 2*x + 1)*x^6 + 27*x^7 - 108*sqrt(-3*x^2 - 2*x + 1)*x^5 - 171*x^6 + 135*sqrt(-3*x^2 - 2*x + 1)*x^4 + 375*x^5 + 46*sqrt(-3*x^2 - 2*x + 1)*x^3 - 173*x^4 + 9*sqrt(-3*x^2 - 2*x + 1)*x^2 - 49*x^3 - 6*sqrt(-3*x^2 - 2*x + 1)*x - 15*x^2 + sqrt(-3*x^2 - 2*x + 1) + 7*x - 1))/( 4*(3*x^2 + 2*x - 1)^4*x^2):
  Gser:=series(G, x=0, 30):
  seq(coeff(Gser, x, n), n=0..26);

G.f.: ((x-1+sqrt(-3*x^2 - 2*x+1))*(27*sqrt(-3*x^2 - 2*x + 1)*x^6 + 27*x^7 - 108*sqrt(-3*x^2 - 2*x + 1)*x^5 - 171*x^6 + 135*sqrt(-3*x^2 - 2*x + 1)*x^4 + 375*x^5 + 46*sqrt(-3*x^2 - 2*x + 1)*x^3 - 173*x^4 + 9*sqrt(-3*x^2 - 2*x + 1)*x^2 - 49*x^3 - 6*sqrt(-3*x^2 - 2*x + 1)*x - 15*x^2 + sqrt(-3*x^2 - 2*x + 1) + 7*x - 1))/( 4*(3*x^2 + 2*x - 1)^4*x^2).

D-finite with recurrence -(n+2)*(208042818093439115480236359*n^2 +3624614398456581514732421474*n -17721487814464945136538072251)*a(n) -(n+1)*(208042818093439115480236359*n^2 -41719745257135632687267408740*n +158505784032262104018605336605)*a(n-1) +(-13360215714657466655169907343*n^3 +118659841630751948460172231402*n^2 -123756458774685279991682146443*n -283543805031439122184609156016)*a(n-2) +(118036702571591403784149448443*n^3 -1525771475215968386687916047321*n^2 +4755582466160131138387124654521*n -4142597862823901093548746996315)*a(n-3) +(-176751907767445269010514270775*n^3 +2983064124441697753911146724326*n^2 -14709907226642052191037550511297*n +13263795264370017511434242152362)*a(n-4) +(-272197576813729306989090076649*n^3 +1341914897255725751921825738923*n^2 +2457351063573329630789375171733*n -7112280079323183611739056078799)*a(n-5) +3*(140641582500711132344919452197*n^3 -2013691622157844732623667550670*n^2 +6061316376844627496454860983745*n -5394852349759561206535461042300)*a(n-6) +9*(31431304934630931225275881933*n^3 -175191266960061764283712600119*n^2 +116063361456271209040196525891*n +17631740228382449873765632167)*a(n-7) -54*(n-6) *(1909106552786855250861701283*n^2 -11218051836500565448490163661*n +11426761828186879119687838319)*a(n-8)=0. - R. J. Mathar, Mar 30 2024

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A333498 Sum of the heights of all Motzkin paths of length n.

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A129181 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n such that the area between the x-axis and the path is k (n>=0; 0<=k<=floor(n^2/4)).

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A367778 a(n) is the sum of the squares of the areas under Motzkin paths of length n.

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A367779 a(n) is the sum of the cubed areas under Motzkin paths of length n.

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