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.

Previous Showing 11-18 of 18 results.

A367640 G.f. A(x) satisfies A(x) = (1 + x)^2 + x*A(x)^3 / (1 + x)^2.

Original entry on oeis.org

1, 3, 8, 36, 200, 1220, 7896, 53220, 369528, 2624772, 18981864, 139287588, 1034475624, 7761249476, 58735359032, 447827171556, 3436759851672, 26526255859716, 205782644595912, 1603655203428900, 12548225647402248, 98548826076070596, 776552629964300952
Offset: 0

Views

Author

Seiichi Manyama, Nov 25 2023

Keywords

Crossrefs

Cf. A367639, A367641.
Cf. A366221, A366266.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(2*k+2,n-k) * binomial(3*k,k)/(2*k+1).
D-finite with recurrence 2*n*(14*n+71)*(2*n+1)*a(n) +3*(-150*n^3-209*n^2-379*n+228)*a(n-1) +9*(-30*n^3-981*n^2+4297*n-3624)*a(n-2) +27*(n-4)*(22*n^2-491*n+1151)*a(n-3) +81*(n-4)*(n-5)*(6*n-49)*a(n-4)=0. - R. J. Mathar, Dec 04 2023

A366591 G.f. A(x) satisfies A(x) = 1 + x^3*(1+x)^2*A(x)^3.

Original entry on oeis.org

1, 0, 0, 1, 2, 1, 3, 12, 18, 24, 75, 180, 295, 620, 1612, 3365, 6580, 15365, 35728, 74906, 163099, 379242, 848148, 1848693, 4187193, 9583209, 21417924, 48067371, 109877922, 250010451, 564688551, 1286128272, 2944963788, 6714338592, 15313680087, 35108572386
Offset: 0

Views

Author

Seiichi Manyama, Oct 14 2023

Keywords

Crossrefs

Cf. A366221, A366590, A366592.
Cf. A366555.

Programs

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

Formula

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

A366592 G.f. A(x) satisfies A(x) = 1 + x^4*(1+x)^2*A(x)^3.

Original entry on oeis.org

1, 0, 0, 0, 1, 2, 1, 0, 3, 12, 18, 12, 15, 72, 180, 240, 235, 512, 1552, 3080, 4123, 5810, 13825, 33200, 58813, 85932, 151578, 346920, 726897, 1242234, 2025177, 3952704, 8509875, 16525872, 28565064, 50849280, 102266019, 208932438, 391951131, 699037248, 1313756457
Offset: 0

Views

Author

Seiichi Manyama, Oct 14 2023

Keywords

Crossrefs

Cf. A366221, A366590, A366591.
Cf. A366556.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(2*k, n-4*k)*binomial(3*k, k)/(2*k+1));

Formula

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

A366222 G.f. A(x) satisfies A(x) = 1 + x*(1 + x)^4*A(x)^3.

Original entry on oeis.org

1, 1, 7, 42, 287, 2114, 16338, 130802, 1075355, 9025656, 77021482, 666267502, 5829209046, 51492030953, 458612500526, 4113879873624, 37133888342707, 337041718357465, 3074153880004188, 28162578841220534, 259020296989987934, 2390818256963083305
Offset: 0

Views

Author

Seiichi Manyama, Oct 04 2023

Keywords

Crossrefs

Cf. A364475, A366200, A366221.
Cf. A099235, A360082.

Programs

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

Formula

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

A366239 G.f. A(x) satisfies A(x) = 1 + x + x*(1 + x)^2*A(x)^3.

Original entry on oeis.org

1, 2, 8, 49, 329, 2401, 18452, 147140, 1206157, 10101011, 86047138, 743288984, 6495476548, 57321239999, 510104531479, 4572492374150, 41247768216331, 374175606700172, 3411195598361653, 31236732721224722, 287182875831208468, 2649838553953071239
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2023

Keywords

Crossrefs

Cf. A364336, A366240, A366241.
Cf. A366221.

Programs

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

Formula

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

A366590 G.f. A(x) satisfies A(x) = 1 + x^2*(1+x)^2*A(x)^3.

Original entry on oeis.org

1, 0, 1, 2, 4, 12, 30, 84, 238, 680, 1993, 5882, 17575, 52976, 160870, 491924, 1512940, 4677672, 14529744, 45320640, 141897039, 445792908, 1404899598, 4440113940, 14069493813, 44689897200, 142268117566, 453839997836, 1450547245960, 4644492976232, 14896047099592
Offset: 0

Views

Author

Seiichi Manyama, Oct 14 2023

Keywords

Crossrefs

Cf. A366221, A366591, A366592.
Cf. A019497.

Programs

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

Formula

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

A378786 G.f. A(x) satisfies A(x) = 1 + x * (1+x)^2 * A(x)^4.

Original entry on oeis.org

1, 1, 6, 39, 296, 2435, 21138, 190603, 1767968, 16761424, 161697576, 1582171216, 15664531716, 156637712953, 1579664567130, 16048129755157, 164085811289360, 1687224436103842, 17436287104620980, 181001686332329224, 1886522317836670988, 19734386503541838083
Offset: 0

Views

Author

Seiichi Manyama, Dec 07 2024

Keywords

Crossrefs

Cf. A365178, A366216, A366272.
Cf. A002478, A073155, A366221.
Cf. A002293.

Programs

  • PARI
    a(n, r=1, s=2, t=4, u=0) = 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

a(n) = Sum_{k=0..n} binomial(4*k+1,k) * binomial(2*k,n-k)/(4*k+1) = Sum_{k=0..n} binomial(2*k,n-k) * A002293(k).

A382893 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1+x)^2 * A(x) )^2.

Original entry on oeis.org

1, 2, 11, 60, 365, 2350, 15767, 109048, 771993, 5567066, 40751267, 302018484, 2261763205, 17088919814, 130108591407, 997225521136, 7688232599089, 59581977618098, 463890112373563, 3626778446099756, 28461425971969693, 224114796803735774, 1770236735807921863
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2025

Keywords

Crossrefs

Cf. A073155, A382886.
Cf. A366221.

Programs

  • PARI
    a(n, r=2, s=2, t=3, u=0) = 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 + x * (1+x)^2 * A(x)^(3/2) )^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).
G.f.: B(x)^2, where B(x) is the g.f. of A366221.
Previous Showing 11-18 of 18 results.

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