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 1-10 of 10 results.

A364864 G.f. A(x) satisfies A(x) = 1 + x*A(x)^3 / (1 + x*A(x)^3).

Original entry on oeis.org

1, 1, 2, 4, 6, -1, -58, -304, -1090, -2876, -4216, 9244, 106746, 529962, 1874628, 4669760, 4309742, -35179252, -277928680, -1269921008, -4214431912, -9197175241, 30113526, 128659598896, 822227670866, 3453484223084, 10519017940952, 18490932535144
Offset: 0

Views

Author

Seiichi Manyama, Aug 11 2023

Keywords

Links

Crossrefs

Cf. A291534, A364865, A364866, A365218.
Cf. A001764, A271469, A363982, A364736, A378892.
Cf. A002293, A336538.

Programs

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

Formula

a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(3*n+k+1,n) / (3*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} (-2)^k * binomial(n,k) * binomial(4*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * binomial(3*n,k-1) for n > 0.

A364866 G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 + x*A(x)^5).

Original entry on oeis.org

1, 1, 4, 21, 124, 781, 5120, 34474, 236492, 1644222, 11543644, 81623504, 580104672, 4137414963, 29574658416, 211639869236, 1514729242092, 10832683182538, 77342204972120, 550791674067623, 3908735530965604, 27612614422978557, 193943797650498016
Offset: 0

Views

Author

Seiichi Manyama, Aug 11 2023

Keywords

Comments

a(34) is negative.

Links

Crossrefs

Cf. A291534, A364864, A364865, A365218.
Cf. A002295.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^k*2^(n-k)*binomial(n, k)*binomial(5*n+k+1, n)/(5*n+k+1));

Formula

a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(5*n+k+1,n) / (5*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} (-2)^k * binomial(n,k) * binomial(6*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * binomial(5*n,k-1) for n > 0.

A365218 G.f. satisfies A(x) = 1 + x*A(x)^6 / (1 + x*A(x)^6).

Original entry on oeis.org

1, 1, 5, 34, 265, 2232, 19766, 181300, 1706737, 16392049, 159959240, 1581278838, 15800619070, 159321921844, 1618981274136, 16562211506496, 170426473666497, 1762771226922775, 18316562635133813, 191104193378725552, 2001224271292820200
Offset: 0

Views

Author

Seiichi Manyama, Aug 26 2023

Keywords

Comments

Conjecture: Is a(n)>0 correct? It is correct up to the first 10000 terms.

Links

Crossrefs

Cf. A291534, A364864, A364865, A364866.
Cf. A002296.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(6*n+1, k)*binomial(n-1, n-k))/(6*n+1);

Formula

a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(6*n+k+1,n)/(6*n+k+1).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(6*n+k+1,k) * binomial(n-1,n-k)/(6*n+k+1).
a(n) = (1/(6*n+1)) * Sum_{k=0..n} (-1)^(n-k) * binomial(6*n+1,k) * binomial(n-1,n-k).

A363982 G.f. satisfies A(x) = 1 + x*A(x)^2 / (1 + x*A(x)^3).

Original entry on oeis.org

1, 1, 1, -1, -8, -16, 16, 195, 491, -317, -6293, -18608, 4610, 230385, 780625, 107615, -9028280, -34695607, -17401607, 367885509, 1598468196, 1350497961, -15317000747, -75391402496, -88375867528, 643505487144, 3611942077216, 5376931884971
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Crossrefs

Cf. A291534, A364051, A364764.
Cf. A219537.

Programs

  • PARI
    a(n) = if(n==0, 1, sum(k=0, n-1, (-1)^k*binomial(n, k)*binomial(2*n+k, n-1-k))/n);

Formula

a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(2*n+k,n-1-k) for n > 0.

A364865 G.f. satisfies A(x) = 1 + x*A(x)^4 / (1 + x*A(x)^4).

Original entry on oeis.org

1, 1, 3, 11, 43, 170, 657, 2392, 7675, 17603, -11898, -529678, -4783303, -33099464, -201744488, -1130700432, -5917753701, -28985131575, -131668554663, -540199800203, -1862208441834, -4014999475540, 10784817197302, 222255824910088, 1973412557775753
Offset: 0

Views

Author

Seiichi Manyama, Aug 11 2023

Keywords

Crossrefs

Cf. A291534, A364864, A364866.
Cf. A002294, A336540.

Programs

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

Formula

a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(4*n+k+1,n) / (4*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} (-2)^k * binomial(n,k) * binomial(5*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * binomial(4*n,k-1) for n > 0.

A378920 G.f. A(x) satisfies A(x) = 1 + x*A(x)^6/(1 + x*A(x)^2).

Original entry on oeis.org

1, 1, 5, 38, 339, 3308, 34191, 367844, 4076112, 46204209, 533239820, 6244542391, 74016115926, 886276231388, 10704869669941, 130271156244371, 1595708949486866, 19658780721376791, 243429900033986385, 3028086095940468087, 37821457123957529163, 474145963420441744445
Offset: 0

Views

Author

Seiichi Manyama, Dec 11 2024

Keywords

Crossrefs

Cf. A002294, A349362, A364765, A365218, A378892, A378919.
Cf. A000108, A219537, A291534, A365225.

Programs

  • PARI
    a(n, r=1, s=-1, t=6, u=2) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

Formula

G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^5/(1 + x*A(x)^2)).
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(s*k,n-k)/(t*k+u*(n-k)+r).

A364051 G.f. satisfies A(x) = 1 + x*A(x)^2 / (1 + x*A(x)^5).

Original entry on oeis.org

1, 1, 1, -3, -21, -41, 166, 1460, 3445, -13503, -136721, -364412, 1285021, 14694643, 43144726, -132548857, -1709480698, -5456400119, 14285376285, 209281385564, 720201663662, -1572818128366, -26541960203077, -97918748134874, 173825501585400, 3453517916428141
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Crossrefs

Cf. A291534, A363982, A364764.
Cf. A364740.

Programs

  • PARI
    a(n) = if(n==0, 1, sum(k=0, n-1, (-1)^k*binomial(n, k)*binomial(2*n+3*k, n-1-k))/n);

Formula

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

A364764 G.f. satisfies A(x) = 1 + x*A(x)^2 / (1 + x*A(x)^4).

Original entry on oeis.org

1, 1, 1, -2, -14, -27, 70, 625, 1457, -3541, -37403, -98547, 207098, 2564079, 7448923, -12940485, -190014459, -600991549, 827159379, 14802832468, 50584687754, -52159768068, -1193457862093, -4384199208207, 3090291576246, 98618925147291, 388126462227091
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Crossrefs

Cf. A291534, A363982, A364051.
Cf. A364739.

Programs

  • PARI
    a(n) = if(n==0, 1, sum(k=0, n-1, (-1)^k*binomial(n, k)*binomial(2*n+2*k, n-1-k))/n);

Formula

a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(2*n+2*k,n-1-k) for n > 0.

A369190 Expansion of (1/x) * Series_Reversion( x / ((1-x)^2 * (1+x)^4) ).

Original entry on oeis.org

1, 2, 3, -2, -39, -176, -442, -26, 6222, 36062, 113240, 91632, -1303985, -9362520, -34625652, -50327818, 293446186, 2693939308, 11475384425, 23120716658, -62820989127, -813918935104, -3964894957296, -10002153961552, 10192131001136, 250612187843962
Offset: 0

Views

Author

Seiichi Manyama, Feb 10 2024

Keywords

Links

Crossrefs

Cf. A291534, A370107.
Cf. A368467.

Programs

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

Formula

G.f.: exp( Sum_{k>=1} A368467(k) * x^k/k ).
a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(2*(n+1),k) * binomial(4*(n+1),n-k).
a(n) = (1/(n+1)) * [x^n] ( (1-x)^2 * (1+x)^4 )^(n+1).

A370107 Expansion of (1/x) * Series_Reversion( x / ((1-x)^2 * (1+x)^3) ).

Original entry on oeis.org

1, 1, -1, -7, -10, 27, 152, 169, -949, -4286, -2646, 36499, 133684, -376, -1458768, -4325495, 3422105, 59242995, 139491393, -260949134, -2414487452, -4307455022, 15274866472, 97910544003, 119082795965, -805538039024, -3921641157424, -2408010178616, 40104318820288
Offset: 0

Views

Author

Seiichi Manyama, Feb 10 2024

Keywords

Links

Crossrefs

Cf. A291534, A369190.
Cf. A370106.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^k * binomial(2*(n+1), k)*binomial(3*(n+1), n-k))/(n+1);
  • PARI
    my(x='x+O('x^30)); Vec(serreverse(x/((1-x)^2*(1+x)^3))/x) \\ Michel Marcus, Feb 10 2024

Formula

G.f.: exp( Sum_{k>=1} A370106(k) * x^k/k ).
a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(2*(n+1),k) * binomial(3*(n+1),n-k).
a(n) = (1/(n+1)) * [x^n] ( (1-x)^2 * (1+x)^3 )^(n+1).
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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