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 1-6 of 6 results.

A366051 Expansion of (1/x) * Series_Reversion( x/(1-x+x^3) ).

Original entry on oeis.org

1, -1, 1, 0, -3, 9, -16, 13, 29, -157, 391, -562, -32, 3002, -10373, 20747, -18083, -47941, 271117, -712216, 1066699, 122131, -6464446, 22907125, -46951992, 40883304, 120187926, -679375906, 1809757015, -2731745887, -468147579, 17768126376, -63256877763
Offset: 0

Views

Author

Seiichi Manyama, Sep 27 2023

Keywords

Crossrefs

Cf. A049128, A114997, A366052, A366053.
Cf. A071879.

Programs

  • PARI
    a(n) = sum(k=0, n\3, (-1)^(n-k)*binomial(n+1, k)*binomial(n-k+1, n-3*k))/(n+1);

Formula

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

A366053 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1-x+x^3) ).

Original entry on oeis.org

1, 3, 15, 92, 628, 4579, 34917, 275041, 2220472, 18275896, 152780718, 1293657534, 11072033677, 95629771059, 832460471465, 7296161486583, 64331378963164, 570228657335744, 5078345448484216, 45418278349485960, 407749837317844851, 3673300856466182388
Offset: 0

Views

Author

Seiichi Manyama, Sep 27 2023

Keywords

Crossrefs

Cf. A049128, A114997, A366051, A366052.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(n+1, k)*binomial(4*n-2*k+2, n-3*k))/(n+1);

Formula

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

A369231 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / (1-x+x^3)^2 ).

Original entry on oeis.org

1, 1, 2, 7, 26, 98, 385, 1569, 6556, 27908, 120624, 528030, 2336202, 10430155, 46930285, 212597901, 968833424, 4438398734, 20428750419, 94424634294, 438104297376, 2039690282940, 9526029685218, 44617396906698, 209526541600978, 986339358246758, 4653571637230839
Offset: 0

Views

Author

Seiichi Manyama, Jan 17 2024

Keywords

Links

Crossrefs

Cf. A366052, A369232.
Cf. A369229.

Programs

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

Formula

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

A366084 Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1-x-x^3) ).

Original entry on oeis.org

1, 2, 7, 29, 132, 637, 3199, 16536, 87366, 469556, 2558610, 14100033, 78437805, 439838596, 2483300228, 14103794518, 80517436710, 461768157262, 2658979794811, 15366500638407, 89093023210674, 518064484263918, 3020484579372765, 17653011431832906
Offset: 0

Views

Author

Seiichi Manyama, Sep 28 2023

Keywords

Links

Crossrefs

Cf. A049133, A243157, A366085.
Cf. A366052.

Programs

  • PARI
    a(n) = sum(k=0, n\3, (-1)^k*binomial(n+1, k)*binomial(3*n-2*k+1, n-3*k))/(n+1);

Formula

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

A369232 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / (1-x+x^3)^3 ).

Original entry on oeis.org

1, 0, 0, 3, 3, 3, 33, 72, 120, 583, 1731, 3888, 13759, 44775, 119793, 381220, 1250328, 3682284, 11455153, 37174428, 114947724, 359381467, 1157319135, 3663615552, 11581104121, 37220909916, 119192219799, 380580143110, 1225279436706, 3948906772872, 12705801908002
Offset: 0

Views

Author

Seiichi Manyama, Jan 17 2024

Keywords

Links

Crossrefs

Cf. A366052, A369231.
Cf. A369081.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1-x)^3/(1-x+x^3)^3)/x)
  • PARI
    a(n, s=3, t=3, u=3) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u-t+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

Formula

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

A372415 Coefficient of x^n in the expansion of ( (1-x+x^3) / (1-x)^3 )^n.

Original entry on oeis.org

1, 2, 10, 59, 366, 2332, 15121, 99276, 657894, 4391438, 29482320, 198865680, 1346655921, 9149295482, 62336961732, 425760311734, 2914151872614, 19983724103726, 137267022656710, 944287970305935, 6504676822047876, 44861522295224400, 309742638630690264
Offset: 0

Views

Author

Seiichi Manyama, Apr 29 2024

Keywords

Crossrefs

Cf. A372413, A372414.
Cf. A366052.

Programs

  • PARI
    a(n, s=3, t=1, u=3) = sum(k=0, n\s, binomial(t*n, k)*binomial((u-t+1)*n-(s-1)*k-1, n-s*k));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(n,k) * binomial(3*n-2*k-1,n-3*k).
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x)^3 / (1-x+x^3) ). See A366052.
Showing 1-6 of 6 results.

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