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.

A368970 Expansion of (1/x) * Series_Reversion( x * (1-x+x^3)^2 ).

Original entry on oeis.org

1, 2, 7, 28, 121, 546, 2531, 11934, 56867, 272580, 1309505, 6285630, 30057195, 142754008, 671062828, 3108766166, 14108600499, 62170980416, 262108536781, 1027886900446, 3509371721163, 8204350476210, -12172347463045, -361684831407060, -3497893818262311
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2024

Keywords

Links

Crossrefs

Cf. A063033, A368962.
Cf. A368974, A368976.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x+x^3)^2)/x)
  • PARI
    a(n, s=3, t=2, u=0) = sum(k=0, n\s, (-1)^k*binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n+k+1,k) * binomial(3*n-2*k+1,n-3*k).

A370800 Expansion of (1/x) * Series_Reversion( x/(x+1/(1-x+x^3)) ).

Original entry on oeis.org

1, 2, 5, 14, 41, 120, 337, 855, 1671, 434, -20393, -158032, -885329, -4322580, -19407365, -81796098, -325964629, -1226861808, -4319079961, -13880383674, -38282558205, -72411121618, 65816173987, 1746824677851, 12859713835981, 73356840199948, 369390356474509
Offset: 0

Views

Author

Seiichi Manyama, Mar 02 2024

Keywords

Links

Crossrefs

Cf. A370799, A370801.
Cf. A063033.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/(x+1/(1-x+x^3)))/x)

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * b(k), where g.f. B(x) = Sum_{k>=0} b(k)*x^k satisfies B(x) = (1/x) * Series_Reversion( x*(1-x+x^3) ).

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

A368935 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x+x^3) ).

Original entry on oeis.org

1, 2, 7, 29, 132, 637, 3200, 16554, 87576, 471570, 2575885, 14238003, 79487023, 447540164, 2538352756, 14489355578, 83174465721, 479842193453, 2780625587824, 16178040713569, 94467163314370, 553430174678595, 3251969073086610, 19161172609833540, 113186247571818096
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2024

Keywords

Crossrefs

Cf. A129442, A368936, A368937.
Cf. A063033, A368931.

Programs

  • PARI
    a(n) = sum(k=0, n\3, (-1)^k*binomial(n+k, k)*binomial(3*n-2*k+1, n-3*k))/(n+1);
  • PARI
    my(x='x+O('x^30)); Vec(serreverse(x*(1-x)*(1-x+x^3))/x) \\ Michel Marcus, Jan 10 2024

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (-1)^k * binomial(n+k,k) * binomial(3*n-2*k+1,n-3*k).
Showing 1-4 of 4 results.

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