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A366046
Expansion of (1/x) * Series_Reversion( x*(1-x+x^5) ).
2
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
0,3
COMMENTS
a(32) is negative.
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
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
PROG
(PARI) a(n) = sum(k=0, n\5, (-1)^k*binomial(n+k, k)*binomial(2*n-4*k, n-5*k))/(n+1);
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Sep 27 2023
STATUS
approved