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).

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).

A368967 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 * (1-x-x^2)^2 ).

Original entry on oeis.org

1, 4, 28, 238, 2244, 22568, 237199, 2574276, 28627224, 324503718, 3735672880, 43555658640, 513277420803, 6103767231712, 73153216133600, 882708243017414, 10714917867247020, 130752597362068496, 1603069096165788706, 19737123968746454284, 243930175282166574432
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2024

Keywords

Links

Crossrefs

Cf. A368961, A368965.
Cf. A368968.

Programs

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

A368976 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 * (1-x+x^3)^2 ).

Original entry on oeis.org

1, 4, 26, 202, 1729, 15730, 149249, 1460300, 14627340, 149254996, 1545959720, 16212144520, 171789072036, 1836515799464, 19783708310984, 214539449634588, 2340148164406642, 25658221358522584, 282627226176802000, 3126081536554547488, 34706443838025828198
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2024

Keywords

Links

Crossrefs

Cf. A368970, A368974.
Cf. A368968.

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, (-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(5*n-2*k+3,n-3*k).

A372461 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, 370, 4012, 44814, 510198, 5886206, 68579020, 805045276, 9507007686, 112817021332, 1344160003030, 16069300956726, 192662610805386, 2315694030560640, 27893938099222316, 336643301659031102, 4069747367955175236, 49274614400855690158
Offset: 0

Views

Author

Seiichi Manyama, May 01 2024

Keywords

Crossrefs

Cf. A368968, A372465.

Programs

  • PARI
    a(n, s=3, 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/3)} 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 A368968.

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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