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Expansion of (1/x) * Series_Reversion( x*(1-x+x^5) ).
2

%I #13 Dec 04 2023 06:02:21

%S 1,1,2,5,14,41,124,384,1210,3861,12434,40313,131332,429250,1405696,

%T 4606898,15093714,49386035,161204470,524361475,1697564726,5461804480,

%U 17433977340,55085418075,171777442668,526480895241,1576234101044,4565064570082,12573573588000

%N Expansion of (1/x) * Series_Reversion( x*(1-x+x^5) ).

%C a(32) is negative.

%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/5)} (-1)^k * binomial(n+k,k) * binomial(2*n-4*k,n-5*k).

%F D-finite with recurrence

%F +2869*n*(n-1)*(n-2)*(n-3) *(1677311589006610608643886320559970*n

%F -7901147144447740888530692468785127)*(n+1)*a(n)

%F +n*(n-1)*(n-2)*(n-3)

%F *(4812206948859965836199309853686553930*n^2

%F -175013553719393167658676882522877604813*n

%F +722425524622711754521906472526821274049)*a(n-1)

%F -6*(n-1)*(n-2)*(n-3)*(38041469564276713074625931629796582292*n^3

%F -434187019812974222921305047255132800148*n^2

%F +1511627766181757985191668395762224462787*n

%F -1439281919744399515865257001890323358373)*a(n-2)

%F +24*(n-2)*(n-3)*(23107055333611559369905978901014910472*n^4

%F -291637186969535206075427515674585653736*n^3

%F +1307639647775331737625407609014469136258*n^2

%F -2417805672147912309219658141920321176114*n

%F +1512007871663508796078252300169686470055)*a(n-3)

%F -1440*(n-3)*(84804544319929041737751787189252800*n^5

%F -1067895117008250068418057111395610000*n^4

%F +3937834774286868181364955550730022660*n^3

%F +975312620367454094109649406073471780*n^2

%F -32021390554042442142065879328318104181*n

%F +47001806684644394483446146792519879754)*a(n-4)

%F +72*(755178462485403935795686391926983696*n^6

%F -9721973068673624889003370906133735808*n^5

%F +28265101220259707812286453812712428560*n^4

%F +142279853462074595032386388289908608780*n^3

%F -1109234048552890437383368746114907399821*n^2

%F +2608361246800778163937859213150591740973*n

%F -2164380627302236226723222549578816128130)*a(n-5)

%F +48600*(6*n-31)*(3*n-13)*(759087266352800004971495991151992*n^4

%F -9778945772952612092782558107378828*n^3

%F +46005785870710778199033560834476886*n^2

%F -93708439282239876819273711147715309*n

%F +69918682390077087204827334331319595)*a(n-6)

%F +139968*(6*n-37)*(3*n-16)*(2*n-11)*(888737373518089148784593470818*n

%F -3184979270877227150713537195033)*(3*n-14)*(6*n-29)*a(n-7)=0. # _R. J. Mathar_, Dec 04 2023

%p A366046 := proc(n)

%p add((-1)^k * binomial(n+k,k) * binomial(2*n-4*k,n-5*k),k=0..floor(n/5)) ;

%p %/(n+1) ;

%p end proc:

%p seq(A366046(n),n=0..70) ; # _R. J. Mathar_, Dec 04 2023

%o (PARI) a(n) = sum(k=0, n\5, (-1)^k*binomial(n+k, k)*binomial(2*n-4*k, n-5*k))/(n+1);

%Y Cf. A063028, A063033.

%Y Cf. A276989, A366024.

%K sign

%O 0,3

%A _Seiichi Manyama_, Sep 27 2023