A366046 Expansion of (1/x) * Series_Reversion( x*(1-x+x^5) ).

1, 1, 2, 5, 14, 41, 124, 384, 1210, 3861, 12434, 40313, 131332, 429250, 1405696, 4606898, 15093714, 49386035, 161204470, 524361475, 1697564726, 5461804480, 17433977340, 55085418075, 171777442668, 526480895241, 1576234101044, 4565064570082, 12573573588000

Offset: 0

Author

Seiichi Manyama, Sep 27 2023

Keywords

• sign

Comments

a(32) is negative.

Crossrefs

Cf. A063028, A063033.

Cf. A276989, A366024.

Programs

Maple

A366046 := proc(n)
    add((-1)^k * binomial(n+k,k) * binomial(2*n-4*k,n-5*k),k=0..floor(n/5)) ;
    %/(n+1) ;
end proc:
seq(A366046(n),n=0..70) ; # R. J. Mathar, Dec 04 2023

PARI

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

Formula

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).

D-finite with recurrence

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

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

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

*(4812206948859965836199309853686553930*n^2

-175013553719393167658676882522877604813*n

+722425524622711754521906472526821274049)*a(n-1)

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

-434187019812974222921305047255132800148*n^2

+1511627766181757985191668395762224462787*n

-1439281919744399515865257001890323358373)*a(n-2)

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

-291637186969535206075427515674585653736*n^3

+1307639647775331737625407609014469136258*n^2

-2417805672147912309219658141920321176114*n

+1512007871663508796078252300169686470055)*a(n-3)

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

-1067895117008250068418057111395610000*n^4

+3937834774286868181364955550730022660*n^3

+975312620367454094109649406073471780*n^2

-32021390554042442142065879328318104181*n

+47001806684644394483446146792519879754)*a(n-4)

+72*(755178462485403935795686391926983696*n^6

-9721973068673624889003370906133735808*n^5

+28265101220259707812286453812712428560*n^4

+142279853462074595032386388289908608780*n^3

-1109234048552890437383368746114907399821*n^2

+2608361246800778163937859213150591740973*n

-2164380627302236226723222549578816128130)*a(n-5)

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

-9778945772952612092782558107378828*n^3

+46005785870710778199033560834476886*n^2

-93708439282239876819273711147715309*n

+69918682390077087204827334331319595)*a(n-6)

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

-3184979270877227150713537195033)*(3*n-14)*(6*n-29)*a(n-7)=0. # R. J. Mathar, Dec 04 2023