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

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

Author

AJ Bu, Nov 29 2023

Keywords

• nonn

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

• 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(-(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 [phone]*n [phone])*a(n-2) +3*(-641100693*n^2 [phone]*n -5322233482)*a(n-3) +(4162359143*n^2 -33175360881*n [phone])*a(n-4) +3*(1437180249*n^2 -9681487559*n [phone])*a(n-5) +9*(-754462425*n^2 [phone]*n -14939114852)*a(n-6) -27*(218140823*n -693079002)*(n-5)*a(n-7)=0. - R. J. Mathar, Jan 11 2024