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.

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A302550 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} (1 + x^(k*j))^j).

Original entry on oeis.org

1, 3, 6, 11, 17, 36, 50, 94, 148, 254, 386, 671, 1005, 1651, 2543, 4034, 6112, 9599, 14410, 22178, 33189, 50196, 74485, 111591, 164149, 242967, 355317, 520817, 755895, 1099219, 1584520, 2285960, 3275667, 4691845, 6682765, 9512213, 13471240, 19059192, 26851931, 37778822
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 20 2018

Keywords

Comments

Inverse Moebius transform of A026007.

Links

Crossrefs

Cf. A026007, A047966, A047968, A300276, A302549.

Programs

  • Maple
    with(numtheory):
b:= proc(n) option remember;
      add((-1)^(n/d+1)*d^2, d=divisors(n))
    end:
g:= proc(n) option remember;
      `if`(n=0, 1, add(b(k)*g(n-k), k=1..n)/n)
    end:
a:= n-> add(g(d), d=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Jun 21 2018
  • Mathematica
    nmax = 40; Rest[CoefficientList[Series[Sum[-1 + Product[(1 + x^(k j))^j, {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]]
b[n_] := b[n] = SeriesCoefficient[Product[(1 + x^k)^k , {k, 1, n}], {x, 0, n}]; a[n_] := a[n] = SeriesCoefficient[Sum[b[k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}]; Table[a[n], {n, 40}]
b[0] = 1; b[n_] := b[n] = Sum[Sum[(-1)^(j/d + 1) d^2, {d, Divisors[j]}] b[n - j], {j, n}]/n; a[n_] := a[n] = Sum[b[d], {d, Divisors[n]}]; Table[a[n], {n, 40}]

Formula

G.f.: Sum_{k>=1} A026007(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} A026007(d).

A323765 Dirichlet convolution of the integer partition numbers A000041 with the strict partition numbers A000009.

Original entry on oeis.org

1, 1, 3, 5, 9, 10, 22, 20, 37, 44, 65, 68, 127, 119, 182, 226, 307, 335, 511, 544, 782, 913, 1171, 1359, 1908, 2121, 2738, 3286, 4174, 4821, 6305, 7182, 9108, 10739, 13195, 15548, 19465, 22397, 27477, 32423, 39448, 45843, 55995, 64871, 78343, 91761, 109325
Offset: 0

Views

Author

Gus Wiseman, Jan 27 2019

Keywords

Comments

Also the number of strict multiset partitions of constant multiset partitions of integer partitions of n.

Examples

			The a(1) = 1 through a(5) = 10 strict multiset partitions of constant multiset partitions of integer partitions:
  ((1))  ((2))     ((3))          ((4))             ((5))
         ((11))    ((21))         ((31))            ((41))
         ((1)(1))  ((111))        ((22))            ((32))
                   ((1)(1)(1))    ((211))           ((311))
                   ((1))((1)(1))  ((1111))          ((221))
                                  ((2)(2))          ((2111))
                                  ((11)(11))        ((11111))
                                  ((1)(1)(1)(1))    ((1)(1)(1)(1)(1))
                                  ((1))((1)(1)(1))  ((1))((1)(1)(1)(1))
                                                    ((1)(1))((1)(1)(1))

Links

Crossrefs

Cf. A000009, A000041, A001970, A034729, A047968, A050343, A316980, A319066, A323764, A323766, A323774.

Programs

  • Mathematica
    Join[{1}, Table[Sum[PartitionsQ[d]*PartitionsP[n/d],{d,Divisors[n]}],{n,1,100}]]

Formula

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)). - Vaclav Kotesovec, Jan 28 2019

A329439 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} 1 / (1 - x^(k*j^2))).

Original entry on oeis.org

1, 2, 2, 4, 3, 5, 3, 7, 6, 8, 5, 12, 7, 10, 10, 15, 10, 19, 11, 22, 17, 20, 15, 31, 22, 28, 27, 35, 27, 44, 29, 46, 40, 48, 43, 69, 47, 61, 58, 80, 61, 89, 67, 93, 92, 97, 85, 129, 101, 131, 118, 146, 125, 172, 142, 182, 166, 191, 170, 241, 193, 231, 230
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 13 2019

Keywords

Comments

Inverse Moebius transform of A001156.

Crossrefs

Cf. A001156, A047966, A047968.

Programs

  • Mathematica
    nmax = 63; CoefficientList[Series[Sum[-1 + Product[1/(1 - x^(k j^2)), {j, 1, Floor[nmax^(1/2)] + 1}], {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=1} A001156(k) * x^k / (1 - x^k).
a(n) = Sum_{d|n} A001156(d).

A182720 Triangle read by rows: T(n,k) = A000041(k) if k divides n, T(n,k)=0 otherwise.

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 2, 0, 5, 1, 0, 0, 0, 7, 1, 2, 3, 0, 0, 11, 1, 0, 0, 0, 0, 0, 15, 1, 2, 0, 5, 0, 0, 0, 22, 1, 0, 3, 0, 0, 0, 0, 0, 30, 1, 2, 0, 0, 7, 0, 0, 0, 0, 42, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56, 1, 2, 3, 5, 0, 11, 0, 0, 0, 0, 0, 77, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 101, 1, 2, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 135, 1, 0, 3, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 176
Offset: 1

Views

Author

Omar E. Pol, Nov 28 2010

Keywords

Examples

			1,
1, 2,
1, 0, 3,
1, 2, 0, 5,
1, 0, 0, 0, 7,
1, 2, 3, 0, 0, 11,
1, 0, 0, 0, 0, 0, 15,
1, 2, 0, 5, 0, 0, 0, 22,
1, 0, 3, 0, 0, 0, 0, 0, 30,
1, 2, 0, 0, 7, 0, 0, 0, 0, 42

Crossrefs

Cf. A000005, A000041, A051731, A168016, A168017, A168018, A168021. Positive integers of row n give A168017.
Row sums give A047968.

Programs

  • Maple
    A182720 := proc(n,k) if n mod k = 0 then combinat[numbpart](k); else 0; end if ; end proc:
seq(seq(A182720(n,k),k=1..n),n=1..15) ;

Formula

T(n,k) = A051731(n,k)*A000041(k).

A318025 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} 1/(1 - j*x^(k*j))).

Original entry on oeis.org

1, 4, 7, 18, 26, 66, 98, 216, 361, 701, 1171, 2287, 3763, 6887, 11707, 20740, 34637, 60678, 100581, 172609, 285924, 481671, 791317, 1323831, 2156856, 3561119, 5784021, 9459559, 15250217, 24783964, 39713789, 64032664, 102200203, 163617694, 259745174, 413886941, 653715969, 1035539948
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 23 2018

Keywords

Comments

Inverse Moebius transform of A006906.

Links

Crossrefs

Cf. A006906, A047968, A300277, A302549, A318290.

Programs

  • Mathematica
    nmax = 38; Rest[CoefficientList[Series[Sum[-1 + Product[1/(1 - j x^(k j)), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]]
b[n_] := b[n] = SeriesCoefficient[Product[1/(1 - k x^k), {k, 1, n}], {x, 0, n}]; a[n_] := a[n] = SeriesCoefficient[Sum[b[k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}]; Table[a[n], {n, 38}]
Table[Sum[Total[Times @@@ IntegerPartitions[d]], {d, Divisors[n]}], {n, 38}]

Formula

G.f.: Sum_{k>=1} A006906(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} A006906(d).

A329435 Expansion of Sum_{k>=1} (-1 + Product_{j>=2} 1 / (1 - x^(k*j))).

Original entry on oeis.org

0, 1, 1, 3, 2, 6, 4, 10, 9, 15, 14, 29, 24, 39, 44, 65, 66, 102, 105, 154, 170, 225, 253, 356, 385, 503, 583, 749, 847, 1100, 1238, 1572, 1809, 2234, 2579, 3219, 3660, 4484, 5195, 6314, 7245, 8800, 10087, 12141, 14011, 16678, 19196, 22930, 26256, 31099, 35784
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 13 2019

Keywords

Comments

Inverse Moebius transform of A002865.

Crossrefs

Cf. A002865, A047966, A047968, A329436.

Programs

  • Mathematica
    nmax = 51; CoefficientList[Series[Sum[-1 + Product[1/(1 - x^(k j)), {j, 2, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=1} A002865(k) * x^k / (1 - x^k).
a(n) = Sum_{d|n} A002865(d).

A329437 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} 1 / (1 - x^(k*prime(j)))).

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 3, 5, 5, 8, 6, 12, 9, 14, 15, 19, 17, 27, 23, 35, 34, 42, 40, 61, 54, 70, 72, 92, 87, 121, 111, 143, 147, 175, 180, 232, 219, 268, 282, 340, 336, 419, 413, 499, 523, 598, 614, 752, 747, 879, 917, 1058, 1083, 1280, 1306, 1515, 1576, 1783, 1850
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 13 2019

Keywords

Comments

Inverse Moebius transform of A000607.

Crossrefs

Cf. A000607, A047966, A047968, A329438.

Programs

  • Mathematica
    nmax = 59; CoefficientList[Series[Sum[-1 + Product[1/(1 - x^(k Prime[j])), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=1} A000607(k) * x^k / (1 - x^k).
a(n) = Sum_{d|n} A000607(d).

A329438 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} (1 + x^(k*prime(j)))).

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 2, 2, 5, 1, 4, 2, 5, 5, 5, 2, 7, 3, 9, 7, 6, 5, 10, 7, 9, 7, 11, 7, 14, 9, 12, 11, 12, 13, 20, 11, 15, 16, 22, 14, 25, 15, 23, 22, 24, 19, 33, 23, 33, 25, 34, 26, 39, 33, 41, 36, 40, 35, 57, 39, 50, 50, 56, 49, 66, 50, 65, 61, 75, 61
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 13 2019

Keywords

Comments

Inverse Moebius transform of A000586.

Crossrefs

Cf. A000586, A047966, A047968, A329437.

Programs

  • Mathematica
    nmax = 71; CoefficientList[Series[Sum[-1 + Product[(1 + x^(k Prime[j])), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=1} A000586(k) * x^k / (1 - x^k).
a(n) = Sum_{d|n} A000586(d).

A329462 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} (1 + x^(k*j^2))).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 1, 2, 2, 2, 2, 3, 2, 2, 1, 5, 2, 1, 1, 2, 4, 4, 2, 3, 3, 5, 1, 3, 1, 3, 3, 4, 2, 2, 3, 6, 3, 4, 1, 2, 5, 3, 1, 3, 3, 8, 3, 6, 3, 4, 3, 4, 2, 4, 2, 7, 3, 4, 4, 4, 7, 4, 1, 5, 3, 7, 2, 4, 2, 6, 7, 3, 3, 9, 3, 8, 5, 5, 2, 7, 6, 4, 5, 3, 4, 14
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 13 2019

Keywords

Comments

Inverse Moebius transform of A033461.

Crossrefs

Cf. A033461, A047966, A047968, A329439.

Programs

  • Mathematica
    nmax = 90; CoefficientList[Series[Sum[-1 + Product[(1 + x^(k j^2)), {j, 1, Floor[nmax^(1/2)] + 1}], {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=1} A033461(k) * x^k / (1 - x^k).
a(n) = Sum_{d|n} A033461(d).

A329465 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} 1 / (1 - x^(k*j*(j + 1)/2))).

Original entry on oeis.org

1, 2, 3, 4, 3, 8, 5, 8, 9, 11, 8, 20, 12, 17, 20, 25, 18, 36, 25, 38, 39, 44, 37, 68, 51, 63, 69, 85, 69, 113, 90, 117, 117, 136, 128, 189, 154, 185, 195, 239, 206, 288, 253, 308, 321, 358, 333, 457, 406, 476, 485, 566, 521, 671, 629, 734, 737, 833, 794, 1019
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 13 2019

Keywords

Comments

Inverse Moebius transform of A007294.

Crossrefs

Cf. A007294, A047966, A047968, A329439, A329466.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Sum[-1 + Product[1/(1 - x^(k j (j + 1)/2)), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=1} A007294(k) * x^k / (1 - x^k).
a(n) = Sum_{d|n} A007294(d).
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