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 1-6 of 6 results.

A369013 Expansion of (1/x) * Series_Reversion( x * (1-x^2/(1-x))^3 ).

Original entry on oeis.org

1, 0, 3, 3, 27, 60, 355, 1128, 5694, 21610, 102462, 426465, 1978547, 8659386, 40003167, 180241995, 834994605, 3830870574, 17841265598, 82854767805, 388124777739, 1818343250570, 8565240659274, 40398758877564, 191254160050512, 906956708168838, 4312790630717025
Offset: 0

Views

Author

Seiichi Manyama, Jan 11 2024

Keywords

Comments

Satisfies a 7-term D-finite recurrence with 7-order polynomials. - R. J. Mathar, Jan 25 2024

Links

Crossrefs

Cf. A369012, A369014.

Programs

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

A369014 Expansion of (1/x) * Series_Reversion( x * (1-x^3/(1-x))^3 ).

Original entry on oeis.org

1, 0, 0, 3, 3, 3, 36, 78, 129, 685, 2043, 4554, 17233, 57279, 153045, 509848, 1724739, 5117643, 16445555, 55165536, 173225715, 555899673, 1847495415, 5971507824, 19333284247, 63975307425, 209807070669, 685973054145, 2269660792842, 7501194321663, 24725092907853
Offset: 0

Views

Author

Seiichi Manyama, Jan 11 2024

Keywords

Links

Crossrefs

Cf. A369012, A369013.
Cf. A054514, A369011.

Programs

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

A378610 Expansion of (1/x) * Series_Reversion( x * (1 - x/(1 - x))^4 ).

Original entry on oeis.org

1, 4, 30, 276, 2825, 30884, 353108, 4170500, 50485764, 623084056, 7810707894, 99175174284, 1272856327470, 16486135484248, 215212582153840, 2828658852385572, 37401956484705132, 497174193516767600, 6640063367021736728, 89058042321373540912, 1199031374607501831273
Offset: 0

Views

Author

Seiichi Manyama, Dec 01 2024

Keywords

Links

Crossrefs

Cf. A001003, A211789, A369012.
Cf. A243667, A371486, A378613.

Programs

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

G.f.: exp( Sum_{k>=1} A378613(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] 1/(1 - x/(1 - x))^(4*(n+1)).
a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(4*n+k+3,k) * binomial(n-1,n-k).
G.f.: B(x)^4 where B(x) is the g.f. of A243667.
a(n) = 4 * Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(4*n+k+4,n)/(4*n+k+4).

A378612 a(n) = Sum_{k=0..n} binomial(3*n+k-1,k) * binomial(n-1,n-k).

Original entry on oeis.org

1, 3, 27, 264, 2703, 28443, 304740, 3306852, 36225519, 399755001, 4437142467, 49485052224, 554059164036, 6224177431332, 70120015345512, 791898021185484, 8962485528377583, 101626868754849381, 1154295872365035537, 13130360954151723480, 149562006735075309783
Offset: 0

Views

Author

Seiichi Manyama, Dec 01 2024

Keywords

Crossrefs

Cf. A002002, A378611, A378613.
Cf. A369012.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(3*n+k-1, k)*binomial(n-1, n-k));

Formula

a(n) = [x^n] 1/(1 - x/(1 - x))^(3*n).
a(n) = (1/4)^n * [x^(3*n)] 3/(1 - x/(1 - x))^n for n > 0.

A378686 G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(7/3)/(1 - x*A(x)) )^3.

Original entry on oeis.org

1, 3, 27, 313, 4122, 58584, 875897, 13577139, 216224616, 3516601243, 58160887857, 975211608399, 16539799297342, 283243124783136, 4890858070498203, 85060240453556192, 1488653675438168001, 26197808077514204832, 463311206395709908936, 8229849868810254813378
Offset: 0

Views

Author

Seiichi Manyama, Dec 04 2024

Keywords

Crossrefs

Cf. A108447, A371581.
Cf. A349017, A369012.
Cf. A378690, A378692, A378693.
Cf. A378685.

Programs

  • PARI
    a(n, r=3, s=1, t=7, u=3) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)^2/(1 - x*A(x)) )^3.
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A378685.
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).

A378670 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)^(3/2)/(1 - x*A(x)^(3/2)) )^2.

Original entry on oeis.org

1, 2, 11, 78, 627, 5432, 49464, 466726, 4522871, 44747874, 450127999, 4589821576, 47333631828, 492836382192, 5173697858508, 54700317431958, 581946708333055, 6225343630256678, 66921440314606905, 722546760572660030, 7832054418695360555, 85198490262065775840
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2024

Keywords

Crossrefs

Cf. A243659, A369012.
Cf. A010683, A211789, A378668.
Cf. A378612.

Programs

  • PARI
    a(n) = 2*sum(k=0, n, 2^k*(-1)^(n-k)*binomial(n, k)*binomial(3*n+k+2, n)/(3*n+k+2));
  • PARI
    a(n, r=2, s=1, t=4, u=3) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

Formula

G.f.: exp( 2/3 * Sum_{k>=1} A378612(k) * x^k/k ).
G.f.: B(x)^2 where B(x) is the g.f. of A243659.
a(n) = 2 * Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(3*n+k+2,n)/(3*n+k+2).
a(n) = 2 * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(n-1,n-k)/(3*n+k+2).
G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^2/(1 - x*A(x)^(3/2)) )^2.
Showing 1-6 of 6 results.

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