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.

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).

A369486 Expansion of (1/x) * Series_Reversion( x / (1-x) * (1-x-x^2)^2 ).

Original entry on oeis.org

1, 1, 4, 15, 67, 314, 1547, 7865, 41004, 217953, 1176832, 6436676, 35587416, 198569471, 1116741601, 6323669519, 36024382515, 206315985386, 1187205083042, 6860598312545, 39797882898452, 231666709974264, 1352813494962672, 7922553881534274, 46520280837291427
Offset: 0

Views

Author

Seiichi Manyama, Jan 24 2024

Keywords

Links

Crossrefs

Cf. A368957, A368961, A368965, A368967.

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

A370619 Coefficient of x^n in the expansion of ( (1-x) / (1-x-x^2) )^(2*n).

Original entry on oeis.org

1, 0, 4, 6, 44, 120, 610, 2114, 9468, 36384, 155644, 626450, 2638994, 10856924, 45565118, 189579786, 796023260, 3333362040, 14022032560, 58960463548, 248542728364, 1048148750060, 4427187324102, 18712146312998, 79177190666034, 335259593600120, 1420797366753600
Offset: 0

Views

Author

Seiichi Manyama, May 01 2024

Keywords

Crossrefs

Cf. A370617, A370618.
Cf. A368957.

Programs

  • PARI
    a(n, s=2, t=2, u=2) = 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/2)} binomial(2*n+k-1,k) * binomial(n-k-1,n-2*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^2)^2 / (1-x)^2 ). See A368957.

A369076 Expansion of (1/x) * Series_Reversion( x * (1+x^2/(1-x))^2 ).

Original entry on oeis.org

1, 0, -2, -2, 9, 24, -37, -240, -2, 2126, 2919, -16052, -50663, 86940, 631995, 19094, -6491463, -9595434, 54443985, 181532910, -317331187, -2426618056, -133151895, 26332109928, 40544827703, -230619508548, -793966990358, 1384746844832, 10960715925621, 881359815524
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2024

Keywords

Links

Crossrefs

Cf. A168491, A369077.
Cf. A368969, A368973, A368975.
Cf. A368957.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1+x^2/(1-x))^2)/x)
  • PARI
    a(n, s=2, 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/2)} (-1)^k * binomial(2*n+k+1,k) * binomial(n-k-1,n-2*k).
Showing 1-4 of 4 results.

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