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 1-5 of 5 results.

A290971 Write x/(1-x) in the form Sum_{j>=1} a(j)*x^j/(1+a(j)*x^j).

Original entry on oeis.org

1, 2, 0, 6, 0, -6, 0, 54, 0, -30, 0, -114, 0, -126, 0, 4470, 0, -294, 0, -5850, 0, -2046, 0, -92418, 0, -8190, 0, -247674, 0, 2010, 0, 30229110, 0, -131070, 0, -8200914, 0, -524286, 0, -362617770, 0, 183162, 0, -354416634, 0, -8388606, 0, -53614489794, 0
Offset: 1

Views

Author

Gus Wiseman, Aug 16 2017

Keywords

Crossrefs

Cf. A048272, A220418, A260685, A273873, A275870, A281145, A289078, A289501, A290261, A290973.

Programs

  • Mathematica
    nn=20;-Solve[Table[Sum[a[n/d]^d,{d,Divisors[n]}]==-1,{n,nn}],Array[a,nn]][[1,All,2]]

Formula

a(n) = -Sum_t (-1)^v(t) where the sum is over all same-trees of weight n (see A281145 for definition) and v(t) is the number of nodes (branchings and leaves) in t.

A295632 Write 1/Product_{n > 1}(1 - 1/n^s) in the form Product_{n > 1}(1 + a(n)/n^s).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1
Offset: 2

Views

Author

Gus Wiseman, Nov 24 2017

Keywords

Comments

First negative entry is a(1024) = -4.

Crossrefs

Cf. A001055, A045778, A050376, A220418, A220420, A273866, A273873, A289501, A290261, A290262, A290971, A290973, A295279, A295635, A295636.

Programs

  • Mathematica
    nn=100;
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Solve[Table[Length[facs[n]]==Sum[Times@@a/@f,{f,Select[facs[n],UnsameQ@@#&]}],{n,2,nn}],Table[a[n],{n,2,nn}]][[1,All,2]]

A301469 Signed recurrence over enriched r-trees: a(n) = 2 * (-1)^n + Sum_y Product_{i in y} a(y) where the sum is over all integer partitions of n - 1.

Original entry on oeis.org

2, -1, 1, 0, 0, 1, 0, 1, 1, 1, 2, 3, 3, 6, 7, 11, 17, 23, 35, 53, 75, 119, 173, 264, 398, 603, 911, 1411, 2114, 3279, 4977, 7696, 11760, 18253, 27909, 43451, 66675, 103945, 160096, 249904, 385876, 603107, 933474, 1461967, 2266384, 3553167, 5521053, 8664117, 13485744
Offset: 0

Views

Author

Gus Wiseman, Mar 21 2018

Keywords

Crossrefs

Cf. A032305, A055277, A093637, A127524, A196545, A220418, A273866, A273873, A289501, A290261, A290973, A301342-A301345, A301364-A301368, A301422, A301462, A301467, A301470.

Programs

  • Mathematica
    a[n_]:=a[n]=2(-1)^n+Sum[Times@@a/@y,{y,IntegerPartitions[n-1]}];
Array[a,30]

Formula

O.g.f.: 2/(1 + x) + x Product_{i > 0} 1/(1 - a(i) x^i).
a(n) = Sum_t 2^k * (-1)^w where the sum is over all enriched r-trees of size n, k is the number of leaves, and w is the sum of leaves.

A295635 Write 2 - Zeta(s) in the form 1/Product_{n > 1}(1 + a(n)/n^s).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 6, 1, 4, 1, 4, 2, 2, 1, 8, 2, 2, 2, 4, 1, 6, 1, 6, 2, 2, 2, 12, 1, 2, 2, 8, 1, 6, 1, 4, 4, 2, 1, 16, 2, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 16, 1, 2, 4, 10, 2, 6, 1, 4, 2, 6, 1, 24, 1, 2, 4, 4, 2, 6, 1, 16, 6, 2, 1, 16, 2, 2
Offset: 2

Views

Author

Gus Wiseman, Nov 24 2017

Keywords

Crossrefs

Cf. A001055, A045778, A050376, A220418, A220420, A273866, A273873, A289501, A290261, A290262, A290971, A290973, A295279, A295632, A295636.

Programs

  • Mathematica
    nn=100;
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
-Solve[Table[-1==Sum[Times@@a/@f,{f,facs[n]}],{n,2,nn}],Table[a[n],{n,2,nn}]][[1,All,2]]

A295636 Write 2 - Zeta(s) in the form Product_{n > 1}(1 - a(n)/n^s).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 8, 1, 2, 2, 4, 1, 6, 1, 6, 2, 2, 2, 8, 1, 2, 2, 8, 1, 6, 1, 4, 4, 2, 1, 16, 1, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 16, 1, 2, 4, 8, 2, 6, 1, 4, 2, 6, 1, 24, 1, 2, 4, 4, 2, 6, 1, 16, 3, 2, 1, 16, 2, 2, 2
Offset: 2

Views

Author

Gus Wiseman, Nov 24 2017

Keywords

Crossrefs

Cf. A001055, A045778, A050376, A220418, A220420, A273866, A273873, A289501, A290261, A290262, A290971, A290973, A295279, A295632, A295635.

Programs

  • Mathematica
    nn=100;
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
-Solve[Table[-1==Sum[Times@@a/@f,{f,Select[facs[n],UnsameQ@@#&]}],{n,2,nn}],Table[a[n],{n,2,nn}]][[1,All,2]]

Formula

a(n) = Sum_t (-1)^(v(t)-1) where the sum is over all strict tree-factorizations of n (see A295279 for definition) and v(t) is the number of nodes (branchings and leaves) in t.
Showing 1-5 of 5 results.

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