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

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

Author

AJ Bu, Nov 29 2023

Keywords

• nonn

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

• 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.

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