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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: AJ Bu

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AJ Bu has authored 3 sequences.

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

Views

Author

AJ Bu, Nov 29 2023

Keywords

Comments

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).

Links

Crossrefs

Cf. A001006, A057585.

Programs

  • Maple
    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);

Formula

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

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

Views

Author

AJ Bu, Nov 29 2023

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

Cf. A001006, A057585.

Programs

  • Maple
    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);
  • PARI
    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

Formula

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

A367780 a(n) is the sum of the squares of the area under Dyck paths of length 2*n.

Original entry on oeis.org

0, 1, 20, 189, 1356, 8426, 47944, 257085, 1321036, 6574190, 31911320, 151841906, 710828600, 3282862644, 14988894992, 67769474077, 303823057164, 1352059744070, 5977826290936, 26277396651558, 114916296684008, 500229317398156, 2168403190878960, 9364025672275634
Offset: 0

Views

Author

AJ Bu, Nov 29 2023

Keywords

Links

Crossrefs

Cf. A000108, A008549.

Programs

  • Maple
    G:= ((-1 + sqrt(-4*x^2 + 1))*(40*x^4 + 14*sqrt(-4*x^2 + 1)*x^2 - 14*x^2 - sqrt(-4*x^2 + 1) + 1))/( 4*(4*x^2 - 1)^3*x^2):  Gser:=series(G, x=0, 41): seq(coeff(Gser, x, 2*n), n=0..19);
  • Mathematica
    G[x_] := ((-1 + Sqrt[-4*x^2 + 1]) * (40*x^4 + 14*Sqrt[-4*x^2 + 1]*x^2 - 14*x^2 - Sqrt[-4*x^2 + 1] + 1)) /  (4*(4*x^2 - 1)^3*x^2); Gser = Series[G[x], {x, 0, 46}]; Table[Coefficient[Gser, x, 2*n], {n, 0, 23}] (* James C. McMahon, Dec 10 2023 *)

Formula

G.f.: ((-1 + sqrt(-4*x^2 + 1))*(40*x^4 + 14*sqrt(-4*x^2 + 1)*x^2 - 14*x^2 - sqrt(-4*x^2 + 1) + 1))/( 4*(4*x^2 - 1)^3*x^2).
D-finite with recurrence -(n+1)*(133*n-262)*a(n) +4*(564*n^2-1229*n+262)*a(n-1) +4*(-2916*n^2+7294*n-2765)*a(n-2) +16*(596*n-553)*(2*n-3)*a(n-3)=0. - R. J. Mathar, Jan 11 2024

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