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 11-19 of 19 results.

A367282 G.f. satisfies A(x) = 1 + x*A(x)^2 * (1 + x*A(x)^2)^2.

Original entry on oeis.org

1, 1, 4, 18, 94, 527, 3108, 18993, 119214, 763997, 4978304, 32883853, 219690066, 1481858835, 10078051830, 69030877581, 475795428158, 3297527987794, 22965847261928, 160649189379029, 1128201207643744, 7951399289858530, 56222323349767666
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2023

Keywords

Crossrefs

Cf. A002293, A073155, A214372, A367283.
Cf. A000108, A367258.
Cf. A367237.

Programs

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

Formula

If g.f. satisfies A(x) = 1 + x*A(x)^t * (1 + x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(s*k,n-k) / (t*k+u*(n-k)+1).

A379039 G.f. A(x) satisfies A(x) = ( (1 + x) * (1 + x*A(x)^2) )^2.

Original entry on oeis.org

1, 4, 22, 172, 1513, 14356, 143228, 1480956, 15728516, 170558634, 1880568650, 21019304814, 237615558790, 2712066792304, 31210387143556, 361738488066632, 4218907281330372, 49476183230651216, 583066018329260673, 6901459436855306662, 82011678696864842013
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2024

Keywords

Crossrefs

Cf. A004798, A073155, A379037.
Cf. A364337, A379038.

Programs

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

Formula

G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A364337.
a(n) = Sum_{k=0..n} binomial(4*k+2,k) * binomial(4*k+2,n-k)/(2*k+1).

A379279 G.f. A(x) satisfies A(x) = ( (1 + x*A(x)^2) * (1 + x*A(x)) )^2.

Original entry on oeis.org

1, 4, 30, 288, 3125, 36490, 447478, 5683186, 74105002, 986302778, 13344661479, 182998935930, 2537838036761, 35530970858236, 501523116910044, 7129275916213606, 101973703002773268, 1466574750062589956, 21194869324964207133, 307642575576365729486, 4482940969372057898247
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2024

Keywords

Crossrefs

Cf. A371675, A379251, A379282.
Cf. A073155, A379280.
Cf. A215715.

Programs

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

Formula

G.f.: B(x)^2 where B(x) is the g.f. of A215715.
a(n) = Sum_{k=0..n} binomial(2*n+2*k+2,k) * binomial(2*n+2*k+2,n-k)/(n+k+1).

A367283 G.f. satisfies A(x) = 1 + x*A(x)^2 * (1 + x*A(x)^3)^2.

Original entry on oeis.org

1, 1, 4, 20, 116, 728, 4818, 33100, 233824, 1687764, 12393520, 92291681, 695325926, 5290359124, 40591599128, 313725215636, 2440203573816, 19087022233906, 150042056387660, 1184734863936672, 9392213303130904, 74728563957003952, 596531545003840160
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2023

Keywords

Crossrefs

Cf. A002293, A073155, A214372, A367282.
Cf. A001764, A137952.

Programs

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

Formula

If g.f. satisfies A(x) = 1 + x*A(x)^t * (1 + x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(s*k,n-k) / (t*k+u*(n-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).

A379037 G.f. A(x) satisfies A(x) = ( (1 + x) * (1 + x*A(x)^(3/2)) )^2.

Original entry on oeis.org

1, 4, 18, 106, 689, 4782, 34707, 260190, 1999168, 15660176, 124596498, 1004110948, 8179379807, 67239070868, 557098881919, 4647368670950, 39001655222787, 329048378867468, 2789241880512898, 23743798316713368, 202894843070927859, 1739775692700850554
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2024

Keywords

Crossrefs

Cf. A004798, A073155, A379039.
Cf. A364336, A366200.

Programs

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

Formula

G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A364336.
a(n) = 2 * Sum_{k=0..n} binomial(3*k+2,k) * binomial(3*k+2,n-k)/(3*k+2).

A382886 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1+x)^2 * A(x) )^3.

Original entry on oeis.org

1, 3, 21, 154, 1248, 10710, 95751, 882297, 8320812, 79927938, 779303829, 7692585186, 76726084742, 772066751871, 7828529324175, 79908510600542, 820435635949686, 8467306916189517, 87791572491261912, 914032693961190414, 9552050623400554164, 100162810727306404897
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2025

Keywords

Crossrefs

Cf. A073155, A382893.
Cf. A378786, A382406.

Programs

  • PARI
    a(n, r=3, 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

G.f. A(x) satisfies A(x) = ( 1 + x * (1+x)^2 * A(x)^(4/3) )^3.
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)^3, where B(x) is the g.f. of A378786.

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.

A336165 G.f. A(x) satisfies: A(x) = 1 + x * ((1 - x) * A(x))^2.

Original entry on oeis.org

1, 1, 0, -2, -4, -3, 6, 26, 46, 22, -128, -455, -748, -149, 2948, 9400, 14254, -1624, -72876, -212988, -294316, 143030, 1889284, 5104273, 6328244, -6017051, -50569884, -126812057, -138104146, 216071703, 1383709740, 3226295732, 2992392698, -7280984690
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 10 2020

Keywords

Crossrefs

Cf. A000108, A006319, A073155, A115399, A256169.

Programs

  • Mathematica
    nmax = 33; CoefficientList[Series[(1 - Sqrt[1 - 4 x (1 - x)^2])/(2 x (1 - x)^2), {x, 0, nmax}], x]
Table[Sum[(-1)^(n - k) Binomial[2 k, n - k] CatalanNumber[k], {k, 0, n}], {n, 0, 33}]

Formula

G.f.: (1 - sqrt(1 - 4*x*(1 - x)^2)) / (2*x*(1 - x)^2).
G.f.: 1 / (1 - x*(1 - x)^2 / (1 - x*(1 - x)^2 / (1 - x*(1 - x)^2 / (1 - ...)))), a continued fraction.
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,n-k) * A000108(k).
a(-1) = 0, a(0) = 1; a(n) = Sum_{k=0..n-1} (a(k) - a(k-1)) * (a(n-k-1) - a(n-k-2)).
D-finite with recurrence (n+1)*a(n) -5*n*a(n-1) +2*(6*n-7)*a(n-2) +2*(-6*n+11)*a(n-3) +2*(2*n-5)*a(n-4)=0. - R. J. Mathar, Mar 06 2022
Previous Showing 11-19 of 19 results.

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