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A367779
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a(n) is the sum of the cubed areas under Motzkin paths of length n.
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0
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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
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OFFSET
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0,4
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COMMENTS
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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).
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LINKS
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FORMULA
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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
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MAPLE
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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);
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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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