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 15 results. Next

A366267 G.f. A(x) satisfies A(x) = 1 + x + x*A(x)^4.

Original entry on oeis.org

1, 2, 8, 56, 448, 3920, 36288, 349440, 3464448, 35125760, 362522624, 3795914240, 40224968704, 430579701760, 4648899846144, 50568103690240, 553632271155200, 6096025799852032, 67464070696927232, 750003531943903232, 8371814935842258944
Offset: 0

Views

Author

Seiichi Manyama, Oct 06 2023

Keywords

Crossrefs

Cf. A025227, A366266, A366268.
Cf. A366272, A366365.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = 1 + x + x*A[x]^4 + O[x]^(nmax+1) // Normal, {nmax+1}];
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 03 2024 *)
  • PARI
    a(n) = sum(k=0, n, binomial(3*k+1, n-k)*binomial(4*k, k)/(3*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(3*k+1,n-k) * binomial(4*k,k)/(3*k+1).
a(n) = A366272(n) + A366272(n-1).
G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366365.

A366273 G.f. A(x) satisfies A(x) = 1 + x*(1 + x)^4*A(x)^5.

Original entry on oeis.org

1, 1, 9, 81, 849, 9681, 116601, 1459809, 18809121, 247782369, 3322209001, 45187029809, 621970864241, 8647376531249, 121261376439641, 1713085987837889, 24358211622230081, 348325689458584769, 5006342381846708681, 72279683684984063249, 1047789195353379807121
Offset: 0

Views

Author

Seiichi Manyama, Oct 06 2023

Keywords

Crossrefs

Cf. A052709, A366221, A366272.
Cf. A366268.

Programs

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

Formula

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

A366495 G.f. A(x) satisfies A(x) = 1 + x*(1+x)^(3/2)*A(x)^(5/2).

Original entry on oeis.org

1, 1, 4, 16, 74, 366, 1900, 10210, 56315, 317005, 1813860, 10518652, 61684208, 365177622, 2179549853, 13100686947, 79232836206, 481821573994, 2944253855746, 18069720545174, 111333779015326, 688399685561554, 4270250156814421, 26567075153764929
Offset: 0

Views

Author

Seiichi Manyama, Oct 11 2023

Keywords

Crossrefs

Cf. A001006, A366221, A366272, A366273, A366496, A366497, A366498, A366499, A366500, A366501.
Cf. A366433.

Programs

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

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366433.
a(n) = Sum_{k=0..n} binomial(3*k/2,n-k) * binomial(5*k/2,k) / (3*k/2+1).

A366496 G.f. A(x) satisfies A(x) = 1 + x*(1+x)^(5/2)*A(x)^(7/2).

Original entry on oeis.org

1, 1, 6, 36, 251, 1891, 15007, 123593, 1046444, 9052330, 79660406, 710879890, 6418000050, 58515227946, 538008396198, 4982752630656, 46442071874398, 435299781856712, 4100411743983559, 38797120485576155, 368561495153257186, 3513923237883474314
Offset: 0

Views

Author

Seiichi Manyama, Oct 11 2023

Keywords

Crossrefs

Cf. A001006, A366221, A366272, A366273, A366495, A366497, A366498, A366499, A366500, A366501.
Cf. A366435.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*k/2, n-k)*binomial(7*k/2, k)/(5*k/2+1));

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366435.
a(n) = Sum_{k=0..n} binomial(5*k/2,n-k) * binomial(7*k/2,k) / (5*k/2+1).

A366497 G.f. A(x) satisfies A(x) = 1 + x*(1+x)^(7/2)*A(x)^(9/2).

Original entry on oeis.org

1, 1, 8, 64, 596, 6028, 64352, 713812, 8146490, 95040886, 1128369960, 13588883712, 165598378308, 2038279921692, 25303322898120, 316443054086214, 3983011314348183, 50418720131975193, 641444450506307160, 8197477211343267688, 105185927879224420064
Offset: 0

Views

Author

Seiichi Manyama, Oct 11 2023

Keywords

Crossrefs

Cf. A001006, A366221, A366272, A366273, A366495, A366496, A366498, A366499, A366500, A366501.
Cf. A366437.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(7*k/2, n-k)*binomial(9*k/2, k)/(7*k/2+1));

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366437.
a(n) = Sum_{k=0..n} binomial(7*k/2,n-k) * binomial(9*k/2,k) / (7*k/2+1).

A366498 G.f. A(x) satisfies A(x) = 1 + x / ((1+x)^(5/2)*A(x)^(3/2)).

Original entry on oeis.org

1, 1, -4, 16, -74, 386, -2180, 12974, -80087, 507887, -3288564, 21649068, -144458484, 974838450, -6641303895, 45615642021, -315530731215, 2196107692119, -15368596890978, 108073850591598, -763293549312084, 5412015893523096, -38508964818580799
Offset: 0

Views

Author

Seiichi Manyama, Oct 11 2023

Keywords

Crossrefs

Cf. A001006, A366221, A366272, A366273, A366495, A366496, A366497, A366499, A366500, A366501.
Cf. A366431.

Programs

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

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366431.
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(n+3*k/2-1,n-k) * binomial(5*k/2-1,k) / (5*k/2-1).

A366499 G.f. A(x) satisfies A(x) = 1 + x / ((1+x)^3*A(x)^2).

Original entry on oeis.org

1, 1, -5, 25, -145, 945, -6641, 49057, -375361, 2948353, -23634049, 192554753, -1589812225, 13272519937, -111850866433, 950220134913, -8129133081601, 69971682467841, -605546841831425, 5265763716550657, -45988028107350017, 403192288488677377
Offset: 0

Views

Author

Seiichi Manyama, Oct 11 2023

Keywords

Crossrefs

Cf. A001006, A366221, A366272, A366273, A366495, A366496, A366497, A366498, A366500, A366501.
Cf. A213282.

Programs

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

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A213282.
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(n+2*k-1,n-k) * binomial(3*k-1,k) / (3*k-1).

A366500 G.f. A(x) satisfies A(x) = 1 + x / ((1+x)^(7/2)*A(x)^(5/2)).

Original entry on oeis.org

1, 1, -6, 36, -251, 1961, -16477, 145307, -1326227, 12420057, -118666032, 1152120806, -11333969511, 112728949041, -1131701419316, 11452480598696, -116702578057106, 1196469605151736, -12332629378843566, 127727907921601146, -1328542834131885506
Offset: 0

Views

Author

Seiichi Manyama, Oct 11 2023

Keywords

Crossrefs

Cf. A001006, A366221, A366272, A366273, A366495, A366496, A366497, A366498, A366499, A366501.
Cf. A366432.

Programs

  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(n+5*k/2-1, n-k)*binomial(7*k/2-1, k)/(7*k/2-1));

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366432.
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(n+5*k/2-1,n-k) * binomial(7*k/2-1,k) / (7*k/2-1).

A366501 G.f. A(x) satisfies A(x) = 1 + x / ((1+x)^4*A(x)^3).

Original entry on oeis.org

1, 1, -7, 49, -399, 3633, -35511, 363937, -3858079, 41951521, -465296487, 5243459409, -59865074223, 690979478481, -8049598938135, 94522387901505, -1117615459764031, 13294669980012865, -158995530738069703, 1910555096402418545, -23056131790988675279
Offset: 0

Views

Author

Seiichi Manyama, Oct 11 2023

Keywords

Crossrefs

Cf. A001006, A366221, A366272, A366273, A366495, A366496, A366497, A366498, A366499, A366500.
Cf. A213336.

Programs

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

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A213336.
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(n+3*k-1,n-k) * binomial(4*k-1,k) / (4*k-1).

A366594 G.f. A(x) satisfies A(x) = 1 + x^3*(1+x)^3*A(x)^4.

Original entry on oeis.org

1, 0, 0, 1, 3, 3, 5, 24, 60, 102, 258, 816, 1992, 4452, 12012, 33617, 84627, 212823, 577361, 1561077, 4063059, 10715009, 29052015, 78235107, 208358693, 560561391, 1522609569, 4120277283, 11129752269, 30240233739, 82441619605, 224488878600, 611770878012
Offset: 0

Views

Author

Seiichi Manyama, Oct 14 2023

Keywords

Crossrefs

Cf. A366272, A366593, A366595.
Cf. A366588, A366591.
Cf. A366557.

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(3*k,n-3*k) * binomial(4*k,k)/(3*k+1).
Showing 1-10 of 15 results. Next

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

Content is available under The OEIS End-User License Agreement.