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A367778
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a(n) is the sum of the squares of the areas under Motzkin paths of length n.
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0
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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
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OFFSET
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1,3
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COMMENTS
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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).
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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);
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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