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 1-4 of 4 results.

A366023 Expansion of (1/x) * Series_Reversion( x*(1-x)*(1+x^4) ).

Original entry on oeis.org

1, 1, 2, 5, 13, 36, 104, 309, 940, 2915, 9184, 29328, 94747, 309180, 1017824, 3376693, 11279274, 37906330, 128085630, 434913555, 1483226921, 5078436800, 17450556480, 60159492600, 208013078910, 721205983737, 2506764055592, 8733076109732, 30489081691750
Offset: 0

Views

Author

Seiichi Manyama, Sep 26 2023

Keywords

Crossrefs

Cf. A063019, A063026, A366024.

Programs

  • Mathematica
    CoefficientList[InverseSeries[Series[x(1-x)(1+x^4),{x,0,29}],x]/x,x]  (* Stefano Spezia, Sep 26 2023 *)
  • PARI
    a(n) = sum(k=0, n\4, (-1)^k*binomial(n+k, n)*binomial(2*n-4*k, n))/(n+1);

Formula

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

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

A366025 Expansion of (1/x) * Series_Reversion( x*(1-x)/(1+x^5) ).

Original entry on oeis.org

1, 1, 2, 5, 14, 43, 139, 465, 1595, 5577, 19804, 71228, 258946, 950030, 3513050, 13079920, 48993149, 184490361, 698020080, 2652192675, 10115878915, 38717526745, 148655862210, 572412768275, 2209969761924, 8553073927858, 33176952295730, 128960722306128
Offset: 0

Views

Author

Seiichi Manyama, Sep 26 2023

Keywords

Crossrefs

Cf. A006318, A052709, A071969, A365268.
Cf. A365760, A366024.

Programs

  • Mathematica
    CoefficientList[InverseSeries[Series[x(1-x)/(1+x^5),{x,0,28}],x]/x,x] (* Stefano Spezia, Sep 26 2023 *)
  • PARI
    a(n) = sum(k=0, n\5, binomial(n-4*k, k)*binomial(2*n-5*k+1, n-4*k)/(2*n-5*k+1));
  • PARI
    Vec(serreverse(x*(1-x)/(1+x^5)+O(x^30))/x) \\ Michel Marcus, Sep 26 2023

Formula

G.f. satisfies A(x) = 1 + x*A(x)^2*(1 + x^4*A(x)^3).
a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k,k) * binomial(2*n-5*k+1,n-4*k)/(2*n-5*k+1) = (1/(n+1)) * Sum_{k=0..floor(n/5)} binomial(n+1,k) * binomial(2*n-5*k,n-5*k).

A366042 Expansion of (1/x) * Series_Reversion( x*(1-x)*(1-x^5) ).

Original entry on oeis.org

1, 1, 2, 5, 14, 43, 139, 465, 1595, 5577, 19805, 71240, 259037, 950590, 3516110, 13095440, 49068051, 184839543, 699607625, 2659276675, 10147039881, 38853068780, 149240187330, 574913637375, 2220609902199, 8598120578442, 33366877654697, 129758691426484
Offset: 0

Views

Author

Seiichi Manyama, Sep 26 2023

Keywords

Crossrefs

Cf. A006013, A063020, A063030, A366041.
Cf. A366024.

Programs

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/5)} binomial(n+k,n) * binomial(2*n-5*k,n).
Showing 1-4 of 4 results.

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