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.

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).

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).

A369077 Expansion of (1/x) * Series_Reversion( x * (1+x^3/(1-x))^2 ).

Original entry on oeis.org

1, 0, 0, -2, -2, -2, 13, 32, 55, -72, -439, -1152, -506, 4870, 20613, 31744, -26392, -313096, -826529, -654362, 3635175, 16431826, 30100349, -15474300, -262654439, -780688624, -756130333, 3013376172, 15711713509, 31584466782, -6090973971, -250819494954
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2024

Keywords

Links

Crossrefs

Cf. A168491, A369076.
Cf. A368970, A368974, A368976.
Cf. A369011.

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, (-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(n-2*k-1,n-3*k).

A372465 Coefficient of x^n in the expansion of 1 / ( (1-x) * (1-x+x^3) )^(2*n).

Original entry on oeis.org

1, 4, 36, 358, 3740, 40194, 439998, 4879326, 54630316, 616194700, 6991215286, 79700776588, 912207989030, 10475536585674, 120641989237890, 1392811194744288, 16114668707519404, 186798818992569818, 2168990381036497812, 25222834639587149890, 293708687012053512870
Offset: 0

Views

Author

Seiichi Manyama, May 01 2024

Keywords

Crossrefs

Cf. A368976, A372461.

Programs

  • PARI
    a(n, s=3, t=2, u=2) = sum(k=0, n\s, (-1)^k*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)} (-1)^k * binomial(2*n+k-1,k) * binomial(5*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)^2 * (1-x+x^3)^2 ). See A368976.
Showing 1-4 of 4 results.

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