cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A368933 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x-x^5) ).

Original entry on oeis.org

1, 2, 7, 30, 143, 729, 3891, 21471, 121505, 701316, 4112751, 24435424, 146773582, 889813460, 5437598036, 33459382065, 207138653334, 1289231982454, 8062548100445, 50637167131635, 319255808742145, 2019867936975125, 12819928874057325, 81603361510347675
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2024

Keywords

Crossrefs

Cf. A236339, A368931, A368932.
Cf. A276989, A366112.

Programs

  • PARI
    a(n) = sum(k=0, n\5, binomial(n+k, k)*binomial(3*n-4*k+1, n-5*k))/(n+1);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)*(1-x-x^5))/x)

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/5)} binomial(n+k,k) * binomial(3*n-4*k+1,n-5*k).

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

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Sep 27 2023

Keywords

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
Showing 1-2 of 2 results.

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