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 1-7 of 7 results.

A366272 G.f. A(x) satisfies A(x) = 1 + x*(1 + x)^3*A(x)^4.

Original entry on oeis.org

1, 1, 7, 49, 399, 3521, 32767, 316673, 3147775, 31977985, 330544639, 3465369601, 36759599103, 393820102657, 4255079743487, 46313023946753, 507319247208447, 5588706552643585, 61875364144283647, 688128167799619585, 7683686768042639359
Offset: 0

Views

Author

Seiichi Manyama, Oct 06 2023

Keywords

Crossrefs

Cf. A052709, A366221, A366273.
Cf. A366267, A366436.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(3*k,n-k) * binomial(4*k,k)/(3*k+1).
G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366436.

A366431 G.f. A(x) satisfies A(x) = 1 + x * (A(x) / (1 - x))^(5/2).

Original entry on oeis.org

1, 1, 5, 25, 135, 775, 4651, 28845, 183450, 1190050, 7844230, 52389678, 353770190, 2411324700, 16568343325, 114639216915, 798076174113, 5586035989185, 39287407321075, 277508001643575, 1967816928168265, 14003018984540741, 99965175670335750
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Partial sums give A366400.
Cf. A006319, A071724, A213282, A213336, A366432, A366433, A366434, A366435, A366436, A366437.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(n+3*k/2-1,n-k) * binomial(5*k/2,k) / (3*k/2+1).

A366432 G.f. A(x) satisfies A(x) = 1 + x * (A(x) / (1 - x))^(7/2).

Original entry on oeis.org

1, 1, 7, 49, 378, 3136, 27363, 247597, 2302511, 21872361, 211336755, 2070577285, 20522662832, 205411356794, 2073258075175, 21078157565623, 215658366319375, 2218853063356937, 22942886758494094, 238284942878492146, 2484736162773443446
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Partial sums give A366401.
Cf. A366431, A366433, A366434, A366435, A366436, A366437.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(n+5*k/2-1,n-k) * binomial(7*k/2,k) / (5*k/2+1).

A366433 G.f. A(x) satisfies A(x) = 1 + x * ((1 - x) / A(x))^(3/2).

Original entry on oeis.org

1, 1, -3, 9, -37, 171, -849, 4421, -23820, 131676, -742616, 4255944, -24714276, 145103426, -859920585, 5137093695, -30902681230, 187034086170, -1138106903928, 6958662440416, -42729903714420, 263400623938140, -1629378251621535, 10111374706286895
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Partial sums give A366404.
Cf. A366431, A366432, A366434, A366435, A366436, A366437.

Programs

  • Mathematica
    Table[(-1)^(n-1) * Sum[Binomial[5*k/2 - 1, k]*Binomial[3*k/2, n - k]/(5*k/2 - 1), {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 10 2023 *)
  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(5*k/2-1, k)*binomial(3*k/2, n-k)/(5*k/2-1));

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(5*k/2-1,k) * binomial(3*k/2,n-k) / (5*k/2-1).
a(n) ~ -(-1)^n * sqrt(4*10^(1/3) + 10^(2/3) - 5) * 3^(n + 1/2) * 5^(n-1) / (sqrt(Pi) * (2 + 10^(1/3)) * n^(3/2) * (4*10^(1/3) + 10^(2/3) - 11)^n). - Vaclav Kotesovec, Oct 10 2023

A366434 G.f. A(x) satisfies A(x) = 1 + x * ((1 - x) / A(x))^2.

Original entry on oeis.org

1, 1, -4, 16, -84, 496, -3140, 20832, -142932, 1005856, -7220100, 52657392, -389088084, 2906551440, -21914464708, 166548194240, -1274531623764, 9812792232768, -75955668337412, 590742300208848, -4614140648464980, 36178872976542768, -284664427193774916
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Partial sums give A366364.
Cf. A366431, A366432, A366433, A366435, A366436, A366437.

Programs

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

Formula

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

A366435 G.f. A(x) satisfies A(x) = 1 + x * ((1 - x) / A(x))^(5/2).

Original entry on oeis.org

1, 1, -5, 25, -160, 1150, -8851, 71345, -594530, 5080300, -44272760, 391961328, -3515490820, 31874449160, -291676084205, 2690284784605, -24985250240043, 233447554879855, -2192862233710505, 20696454624488125, -196168344717398010, 1866499116495323946
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Partial sums give A366405.
Cf. A366431, A366432, A366433, A366434, A366436, A366437.

Programs

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

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(7*k/2-1,k) * binomial(5*k/2,n-k) / (7*k/2-1).

A366437 G.f. A(x) satisfies A(x) = 1 + x * ((1 - x) / A(x))^(7/2).

Original entry on oeis.org

1, 1, -7, 49, -427, 4165, -43435, 473977, -5344333, 61772179, -727993301, 8714701219, -105672771225, 1295237037815, -16021641194545, 199747074505773, -2507395464414008, 31664298046926328, -401994771266030880, 5127701624204157600, -65684716411944207144
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Partial sums give A366406.
Cf. A366431, A366432, A366433, A366434, A366435, A366436.

Programs

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

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(9*k/2-1,k) * binomial(7*k/2,n-k) / (9*k/2-1).
Showing 1-7 of 7 results.

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