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 1-8 of 8 results.

A371523 G.f. A(x) satisfies A(x) = (1 + x*A(x)^3 / (1-x))^2.

Original entry on oeis.org

1, 2, 15, 142, 1533, 17924, 220936, 2827218, 37202580, 500228562, 6842899886, 94931338876, 1332438761910, 18887047322030, 269986427261981, 3887654399820062, 56337997080499605, 821021578186212094, 12024687038651388155, 176900548019426869808, 2612917215947948178941
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Cf. A349333, A371379, A371519, A371521, A371522.
Cf. A270386, A371518.

Programs

  • PARI
    a(n) = 2*sum(k=0, n, binomial(n-1, n-k)*binomial(6*k+1, k)/(5*k+2));

Formula

a(n) = 2 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+1,k)/(5*k+2).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A349333.

A371379 G.f. A(x) satisfies A(x) = (1 + x*A(x)^(3/2) / (1-x))^4.

Original entry on oeis.org

1, 4, 34, 344, 3859, 46240, 579722, 7511272, 99782617, 1351784792, 18604380884, 259395030992, 3656180724752, 52011780756632, 745799171500502, 10768038899631476, 156414710103922340, 2284233700081510820, 33517461646190624690, 493917761019513208800
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Cf. A349333, A371519, A371521, A371522, A371523.

Programs

  • PARI
    a(n) = 4*sum(k=0, n, binomial(n-1, n-k)*binomial(6*k+3, k)/(5*k+4));

Formula

a(n) = 4 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+3,k)/(5*k+4).
G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A349333.

A371517 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^4.

Original entry on oeis.org

1, 4, 26, 188, 1459, 11892, 100444, 871528, 7722557, 69590628, 635807180, 5876094308, 54836925779, 516029817620, 4891147100886, 46653935716492, 447490869463145, 4313492172957396, 41763413498670702, 405968522259130636, 3960526930400038404
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Cf. A045868, A371516, A371520, A371521.
Cf. A349331, A371483, A371518.
Cf. A371486.

Programs

  • PARI
    a(n) = 4*sum(k=0, n, binomial(n-1, n-k)*binomial(4*k+3, k)/(3*k+4));

Formula

a(n) = 4 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(4*k+3,k)/(3*k+4) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(4*k+4,k)/(k+1).
G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A349331.

A371516 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^3.

Original entry on oeis.org

1, 3, 15, 82, 477, 2901, 18235, 117555, 773085, 5166478, 34987170, 239570655, 1655933060, 11538839130, 80971109712, 571702698185, 4058556404958, 28951715755830, 207424064434502, 1491898838023884, 10768487956456506, 77977009814421534, 566310026687320290
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Cf. A045868, A371517, A371520, A371521.
Cf. A270386, A307678.
Cf. A371483.

Programs

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

Formula

a(n) = 3 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(3*k+2,k)/(2*k+3) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(3*k+3,k)/(k+1).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A307678.

A371519 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1-x))^5.

Original entry on oeis.org

1, 5, 45, 470, 5375, 65231, 825225, 10764185, 143739440, 1955340360, 27001732972, 377530388235, 5333865386885, 76031188364860, 1092117166466660, 15792298241897649, 229704197116753825, 3358528175751886765, 49333470827844265285, 727680248026484478405
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Cf. A349333, A371379, A371521, A371522, A371523.
Cf. A270386, A371483, A371486.
Cf. A130564.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n-1, n-k)*binomial(6*k+4, k)/(k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+4,k)/(k+1).
G.f.: A(x) = B(x/(1-x)), where B(x) = (1/x) * Series_Reversion( x*(1-x)^5 ).
G.f.: A(x) = B(x)^5 where B(x) is the g.f. of A349333.

A371520 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^5.

Original entry on oeis.org

1, 5, 40, 360, 3495, 35726, 378965, 4133080, 46059020, 522196465, 6004261226, 69849651025, 820651943130, 9723556336780, 116056250171385, 1394082307995626, 16840510019954835, 204453614350921540, 2493311080293185200, 30528431677508637205, 375155454309681439001
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Cf. A045868, A371516, A371517, A371521.
Cf. A349332, A371486.

Programs

  • PARI
    a(n) = 5*sum(k=0, n, binomial(n-1, n-k)*binomial(5*k+4, k)/(4*k+5));

Formula

a(n) = 5 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(5*k+4,k)/(4*k+5) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(5*k+5,k)/(k+1).
G.f.: A(x) = B(x)^5 where B(x) is the g.f. of A349332.

A371522 G.f. A(x) satisfies A(x) = (1 + x*A(x)^2 / (1-x))^3.

Original entry on oeis.org

1, 3, 24, 235, 2586, 30603, 380359, 4896753, 64731747, 873539236, 11984536632, 166661420814, 2343950447112, 33282048811530, 476462982915993, 6869620848003570, 99663539644072305, 1453861111238442363, 21312207036239313936, 313783619269186619589
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Cf. A349333, A371379, A371519, A371521, A371523.

Programs

  • PARI
    a(n) = 3*sum(k=0, n, binomial(n-1, n-k)*binomial(6*k+2, k)/(5*k+3));

Formula

a(n) = 3 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+2,k)/(5*k+3).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A349333.

A371541 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1+x))^6.

Original entry on oeis.org

1, 6, 45, 410, 4110, 43746, 485237, 5547396, 64901670, 773296320, 9350929395, 114464359296, 1415620823147, 17661466502796, 222017667461685, 2809362871991380, 35755481454362355, 457410181217186886, 5878378983480722222, 75856853080508789406
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Cf. A349362, A371537, A371538, A371539, A371540.
Cf. A371521.

Programs

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

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,n-k) * binomial(6*k+6,k)/(k+1).
G.f.: A(x) = B(x)^6 where B(x) is the g.f. of A349362.
Showing 1-8 of 8 results.

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