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.

A368966 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x-x^3)^2 ).

Original entry on oeis.org

1, 3, 15, 93, 644, 4769, 36953, 295867, 2428373, 20322566, 172759032, 1487632887, 12948891408, 113748663495, 1007117650350, 8978151790011, 80519598139947, 725976573163011, 6576546244337046, 59829384514916820, 546375444906314661, 5006934930385254672
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2024

Keywords

Links

Crossrefs

Cf. A368962, A368968.
Cf. A368965.

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(4*n-2*k+2,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).

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).

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).

A369511 Expansion of (1/x) * Series_Reversion( x * ((1-x)^2-x^3)^2 ).

Original entry on oeis.org

1, 4, 26, 206, 1815, 17082, 168159, 1710234, 17828973, 189504744, 2045971440, 22374997320, 247344411792, 2759394009008, 31027178033064, 351270123392892, 4000793799046578, 45809545263096832, 527010005799822844, 6088666065809281348, 70612995488695876634
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2024

Keywords

Links

Crossrefs

Cf. A368962, A369514.
Cf. A369214.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^2-x^3)^2)/x)
  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n+k+1, k)*binomial(5*n-k+3, n-3*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(2*n+k+1,k) * binomial(5*n-k+3,n-3*k).

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).

A370626 Coefficient of x^n in the expansion of 1 / (1-x-x^3)^(2*n).

Original entry on oeis.org

1, 2, 10, 62, 402, 2662, 17914, 122040, 839154, 5811758, 40482530, 283311470, 1990464450, 14030571258, 99179197512, 702789627712, 4990636603986, 35506061422530, 253030893941362, 1805893735209486, 12906043894108162, 92346511605008562, 661494201448139850
Offset: 0

Views

Author

Seiichi Manyama, May 01 2024

Keywords

Crossrefs

Cf. A368962.

Programs

  • PARI
    a(n, s=3, t=2, u=0) = 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(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-x^3)^2 ). See A368962.
Showing 1-7 of 7 results.

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