cp's OEIS Frontend

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

%I A367779 #20 Mar 30 2024 10:19:57
%S A367779 0,0,1,10,118,818,5092,27564,137836,644836,2870189,12266918,50724954,
%T A367779 204046142,801892081,3089123960,11696423536,43623049688,160547844283,
%U A367779 583940294930,2101624362838,7492542382034,26484322064854,92891831844644,323514376584988,1119432296516028,3850521166068067
%N A367779 a(n) is the sum of the cubed areas under Motzkin paths of length n.
%C A367779 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).
%H A367779 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 A367779 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).
%F A367779 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
%p A367779 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):
%p A367779   Gser:=series(G, x=0, 30):
%p A367779   seq(coeff(Gser, x, n), n=0..26);
%Y A367779 Cf. A001006, A057585.
%K A367779 nonn
%O A367779 0,4
%A A367779 _AJ Bu_, Nov 29 2023