A367778 - OEIS

%I #27 May 21 2024 11:30:03

%S 0,1,6,40,198,910,3848,15492,59920,224917,824074,2960828,10466610,

%T 36498195,125801144,429284612,1452174984,4874940295,16254780970,

%U 53873727516,177594715034,582603630260,1902860189328,6190199896600,20064013907288,64815504118695,208739559416878,670345766842528

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

%C 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).

%H AJ Bu, <a href="https://arxiv.org/abs/2310.17026">Explicit Generating Functions for the Sum of the Areas Under Dyck and Motzkin Paths (and for Their Powers)</a>, arXiv:2310.17026 [math.CO], 2023.

%F 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)).

%F 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

%e 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.

%p 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);

%o (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

%Y Cf. A001006, A057585.

%K nonn

%O 1,3

%A _AJ Bu_, Nov 29 2023