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 1-7 of 7 results.

A368962 Expansion of (1/x) * Series_Reversion( x * (1-x-x^3)^2 ).

Original entry on oeis.org

1, 2, 7, 32, 165, 910, 5251, 31314, 191463, 1193808, 7561825, 48522630, 314752515, 2060587112, 13597183916, 90342651982, 603886553067, 4058197580308, 27401404029181, 185806213609730, 1264774546754103, 8639226724499070, 59198404680049915
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2024

Keywords

Links

Crossrefs

Cf. A368966, A368968.
Cf. A368961.

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, 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)} binomial(2*n+k+1,k) * binomial(3*n-2*k+1,n-3*k).

A368968 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 * (1-x-x^3)^2 ).

Original entry on oeis.org

1, 4, 26, 206, 1813, 17030, 167229, 1695920, 17624932, 186722580, 2009077416, 21894695420, 241170873096, 2680761546396, 30032284769832, 338744791093796, 3843699928567438, 43844993166845920, 502497843180361288, 5783367971991398760, 66815895492710846218
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2024

Keywords

Links

Crossrefs

Cf. A368962, A368966, A369011.
Cf. A368967.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2*(1-x-x^3)^2)/x)
  • PARI
    a(n, s=3, t=2, u=2) = sum(k=0, n\s, 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)} binomial(2*n+k+1,k) * binomial(5*n-2*k+3,n-3*k).

A368965 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x-x^2)^2 ).

Original entry on oeis.org

1, 3, 17, 117, 895, 7309, 62410, 550431, 4975297, 45846977, 429095387, 4067760593, 38977419018, 376901628882, 3673226867356, 36043590216621, 355800292078095, 3530878133357175, 35205183620396571, 352505713454687599, 3543078943592291301
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2024

Keywords

Links

Crossrefs

Cf. A368961, A368967.
Cf. A368966.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)*(1-x-x^2)^2)/x)
  • PARI
    a(n, s=2, t=2, u=1) = sum(k=0, n\s, 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/2)} binomial(2*n+k+1,k) * binomial(4*n-k+2,n-2*k).

A369011 Expansion of (1/x) * Series_Reversion( x * (1-x^3/(1-x))^2 ).

Original entry on oeis.org

1, 0, 0, 2, 2, 2, 17, 36, 59, 240, 669, 1452, 4538, 13574, 34505, 99816, 299112, 825768, 2364715, 7023466, 20182611, 58327250, 172491553, 505163444, 1476966513, 4370772096, 12924382671, 38149522136, 113266357609, 336894290910, 1001473479313, 2985508193930
Offset: 0

Views

Author

Seiichi Manyama, Jan 11 2024

Keywords

Links

Crossrefs

Cf. A211789, A368957.
Cf. A054514, A369014.
Cf. A368962, A368966, A368968.

Programs

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

A368974 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x+x^3)^2 ).

Original entry on oeis.org

1, 3, 15, 89, 580, 4009, 28857, 213967, 1622869, 12531090, 98171544, 778364379, 6233789872, 50355710215, 409790010350, 3356429972859, 27647745771339, 228890532343859, 1903475080613014, 15893483726218904, 133190665385526309, 1119863488613216952
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2024

Keywords

Links

Crossrefs

Cf. A368970, A368976.
Cf. A368966.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)*(1-x+x^3)^2)/x)
  • PARI
    a(n, s=3, t=2, u=1) = 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(4*n-2*k+2,n-3*k).

A372459 Coefficient of x^n in the expansion of 1 / ( (1-x) * (1-x-x^3)^2 )^n.

Original entry on oeis.org

1, 3, 21, 171, 1469, 12988, 116985, 1067545, 9836541, 91313469, 852701256, 8001080244, 75375985841, 712487600698, 6754115819535, 64185511063246, 611287650124125, 5832863405199183, 55750924705841643, 533676328608473118, 5115556211638071944
Offset: 0

Views

Author

Seiichi Manyama, May 01 2024

Keywords

Crossrefs

Cf. A368966.

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(2*n+k-1,k) * binomial(4*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) * (1-x-x^3)^2 ). See A368966.

A369489 Expansion of (1/x) * Series_Reversion( x / (1-x) * (1-x-x^3)^2 ).

Original entry on oeis.org

1, 1, 2, 7, 26, 98, 387, 1589, 6688, 28676, 124880, 550926, 2456831, 11056693, 50152457, 229050621, 1052393802, 4861062466, 22559964766, 105144660498, 491922058878, 2309456782464, 10876596029574, 51372213424194, 243283513468707, 1154929327702775, 5495105429597720
Offset: 0

Views

Author

Seiichi Manyama, Jan 24 2024

Keywords

Links

Crossrefs

Cf. A368962, A368966, A368968, A369011.
Cf. A054514, A369486.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1-x)*(1-x-x^3)^2)/x)
  • PARI
    a(n, s=3, t=2, u=1) = sum(k=0, n\s, 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)} binomial(2*n+k+1,k) * binomial(2*n-2*k,n-3*k).
Showing 1-7 of 7 results.

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