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.

A300862 Solution to 1 = Sum_y Product_{k in y} a(k) for each n > 0, where the sum is over all integer partitions of n with an odd number of parts.

Original entry on oeis.org

1, 1, 0, 0, -1, -1, 0, 1, 1, 0, -2, -3, -2, 2, 7, 6, -3, -15, -19, -2, 32, 54, 24, -64, -153, -123, 95, 389, 444, -43, -966, -1475, -516, 2066, 4414, 3092, -3874, -12480, -12936, 3847, 32445, 45494, 8950, -77282, -147663, -86313, 157456, 435623, 399041, -229616, -1211479, -1535700, -73132
Offset: 1

Views

Author

Gus Wiseman, Mar 13 2018

Keywords

Crossrefs

Cf. A027193, A063834, A220418, A279374, A290261, A290971, A298118, A299202, A299203, A300301, A300436, A300439, A300863, A300864, A300865, A300866.

Programs

  • Mathematica
    a[n_]:=a[n]=1-Sum[Times@@a/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&]}];
Array[a,40]

A300863 Signed recurrence over enriched p-trees: a(n) = (-1)^(n - 1) + Sum_{y1 + ... + yk = n, y1 >= ... >= yk > 0, k > 1} a(y1) * ... * a(yk).

Original entry on oeis.org

1, 0, 2, 2, 6, 14, 34, 82, 214, 566, 1482, 4058, 10950, 30406, 83786, 235714, 658286, 1874254, 5293674, 15189810, 43312542, 125075238, 359185586, 1043712922, 3015569582, 8800146182, 25565402802, 74918274562, 218572345718, 642783954238, 1882606578002
Offset: 1

Views

Author

Gus Wiseman, Mar 13 2018

Keywords

Links

Crossrefs

Cf. A000992, A063834, A099323, A196545, A220418, A273866, A273873, A289501, A290261, A300862, A300864, A300865, A300866.

Programs

  • Mathematica
    a[n_]:=a[n]=(-1)^(n-1)+Sum[Times@@a/@y,{y,Select[IntegerPartitions[n],Length[#]>1&]}];
Array[a,40]

Formula

O.g.f.: (-1/(1+x) + Product 1/(1-a(n)x^n))/2.

A300864 Signed recurrence over strict trees: a(n) = -1 + Sum_{y1 + ... + yk = n, y1 > ... > yk > 0, k > 1} a(y1) * ... * a(yk).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 2, -1, 0, 4, -6, 6, 6, -24, 38, -17, -64, 188, -230, -6, 662, -1432, 1286, 1210, -6362, 10692, -5530, -18274, 57022, -74364, 174, 216703, -489544, 467860, 391258, -2256430, 3948206, -2234064, -6725362, 21920402, -29716570, 2095564, 84595798, -198418242, 197499846
Offset: 1

Views

Author

Gus Wiseman, Mar 13 2018

Keywords

Crossrefs

Cf. A000992, A018819, A063834, A099323, A196545, A220418, A273866, A273873, A289501, A290261, A300862, A300863, A300865, A300866.

Programs

  • Mathematica
    a[n_]:=a[n]=-1+Sum[Times@@a/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&UnsameQ@@#&]}];
-Array[a,40]

A300865 Signed recurrence over binary enriched p-trees: a(n) = (-1)^(n-1) + Sum_{x + y = n, 0 < x <= y < n} a(x) * a(y).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 2, 4, 6, 10, 16, 27, 46, 77, 131, 224, 391, 672, 1180, 2050, 3626, 6344, 11276, 19863, 35479, 62828, 112685, 200462, 360627, 644199, 1162296, 2083572, 3768866, 6777314, 12289160, 22158106, 40255496, 72765144, 132453122, 239936528, 437445448
Offset: 1

Views

Author

Gus Wiseman, Mar 13 2018

Keywords

Crossrefs

Cf. A000992, A001190, A007317, A063834, A099323, A196545, A220418, A273866, A273873, A289501, A290261, A300352, A300442, A300443, A300862, A300863, A300864, A300866.

Programs

  • Mathematica
    a[n_]:=a[n]=(-1)^(n-1)+Sum[a[k]*a[n-k],{k,1,n/2}];
Array[a,50]

A302917 Solution to a(1) = 1 and Sum_y Product_i a(y_i) = 0 for each n > 1, where the sum is over all relatively prime or monic partitions of n.

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 0, 0, 0, 0, -1, 1, -1, -1, 1, 1, -1, -1, 0, 1, 1, -3, 1, 4, -5, -3, 3, 4, 2, -6, -6, 19, -8, -25, 25, 20, -12, -34, 2, 30, 38, -117, 54, 159, -173, -123, 55, 229, 32, -250, -148, 753, -365, -1022, 840, 1121, -847, -1482, -390, 2099
Offset: 1

Views

Author

Gus Wiseman, Apr 15 2018

Keywords

Comments

A relatively prime or monic partition of n is an integer partition of n that is either of length 1 (monic) or whose parts have no common divisor other than 1 (relatively prime).

Crossrefs

Cf. A000837, A036987, A047966, A063834, A093637, A196545, A220418, A289501, A290261, A300486, A300863-A300866, A301462, A302094, A302915, A302916.

Programs

  • Mathematica
    a[n_]:=a[n]=If[n===1,1,0]-Sum[Times@@a/@y,{y,Rest[Select[IntegerPartitions[n],Or[Length[#]===1,GCD@@#===1]&]]}];
Array[a,20]

A305572 a(n) = (-1)^(n-1) + Sum_{d|n, d>1} a(n/d)^d.

Original entry on oeis.org

1, 0, 2, 0, 2, 4, 2, 0, 10, 4, 2, 32, 2, 4, 42, 0, 2, 228, 2, 32, 138, 4, 2, 1536, 34, 4, 1514, 32, 2, 3940, 2, 0, 2058, 4, 162, 102944, 2, 4, 8202, 1536, 2, 51940, 2, 32, 207370, 4, 2, 3538944, 130, 3204, 131082, 32, 2, 15668836, 2082, 1536, 524298, 4, 2, 54327840
Offset: 1

Views

Author

Gus Wiseman, Jun 05 2018

Keywords

Links

Crossrefs

Cf. A196545, A273873, A281145, A289078, A289079, A289501, A290261, A290971, A291441, A300862-A300866.

Programs

  • Mathematica
    a[n_]:=a[n]=(-1)^(n-1)+Sum[a[n/y]^y,{y,Divisors[n]//Rest}];
Array[a,40]
  • PARI
    A305572(n) = ((-1)^(n-1) + sumdiv(n,d,if(d==1,0,A305572(n/d)^d))); \\ Antti Karttunen, Dec 05 2021

Formula

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

A305610 a(n) = (-1)^(n-1) + Sum_{d|n, d>1} binomial(a(n/d) + d - 1, d).

Original entry on oeis.org

1, 0, 2, 0, 2, 3, 2, 0, 6, 3, 2, 11, 2, 3, 12, 0, 2, 38, 2, 11, 14, 3, 2, 90, 8, 3, 68, 11, 2, 127, 2, 0, 18, 3, 16, 1194, 2, 3, 20, 90, 2, 173, 2, 11, 644, 3, 2, 5158, 10, 68, 24, 11, 2, 12762, 20, 90, 26, 3, 2, 12910, 2, 3, 1386, 0, 22, 289, 2, 11, 30, 219, 2
Offset: 1

Views

Author

Gus Wiseman, Jun 06 2018

Keywords

Links

Crossrefs

Cf. A063834, A196545, A220418, A273866, A273873, A275870, A281145, A281146, A289078, A289079, A289501, A290261, A290971, A291441, A300862-A300866.

Programs

  • Mathematica
    a[n_]:=a[n]=(-1)^(n-1)+Sum[Binomial[a[n/d]+d-1,d],{d,Divisors[n]//Rest}];
Array[a,40]
  • PARI
    A305610(n) = ((-1)^(n-1) + sumdiv(n,d,if(d==1,0,binomial(A305610(n/d)+d-1, d)))); \\ Antti Karttunen, Dec 05 2021
Showing 1-7 of 7 results.

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