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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A284828 Expansion of Sum_{i>=2} x^prime(i)/(1 - x^prime(i)) * Product_{j>=i} 1/(1 - x^prime(j)).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 1, 3, 3, 3, 5, 4, 6, 9, 7, 10, 11, 12, 17, 19, 22, 23, 26, 33, 36, 41, 48, 52, 59, 66, 78, 85, 97, 112, 117, 134, 151, 169, 187, 207, 230, 255, 284, 313, 348, 379, 418, 465, 508, 561, 620, 674, 737, 812, 892, 972, 1064, 1157, 1257, 1379, 1503, 1639, 1776, 1935, 2101, 2279, 2483
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 03 2017

Keywords

Comments

Total number of smallest parts in all partitions of n into odd prime parts (A065091).

Examples

			a(16) = 7 because we have [13, 3], [11, 5], [7, 3, 3, 3], [5, 5, 3, 3] and 1 + 1 + 3 + 2 = 7.

Links

Crossrefs

Cf. A065091, A084993, A092268, A092269, A099773, A195820, A281544, A284827.

Programs

  • Mathematica
    nmax = 68; Rest[CoefficientList[Series[Sum[x^Prime[i]/(1 - x^Prime[i]) Product[1/(1 - x^Prime[j]), {j, i, nmax}], {i, 2, nmax}], {x, 0, nmax}], x]]
  • PARI
    x = 'x + O('x ^ 70); concat([0, 0], Vec(sum(i=2, 70, x^prime(i)/(1 - x^prime(i)) * prod(j=i, 70, 1/(1 - x^prime(j)))))) \\ Indranil Ghosh, Apr 05 2017

Formula

G.f.: Sum_{i>=2} x^prime(i)/(1 - x^prime(i)) * Product_{j>=i} 1/(1 - x^prime(j)).

A284829 Expansion of Sum_{i>=1} mu(i)^2*x^i/(1 - x^i) * Product_{j>=i} 1/(1 - mu(j)^2*x^j), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 3, 5, 9, 13, 23, 30, 45, 64, 89, 118, 165, 211, 282, 369, 475, 606, 779, 978, 1236, 1547, 1922, 2375, 2936, 3602, 4403, 5362, 6506, 7864, 9493, 11399, 13661, 16317, 19443, 23122, 27415, 32418, 38268, 45065, 52968, 62125, 72742, 84969, 99112, 115409, 134139, 155665, 180368, 208658, 241051
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 03 2017

Keywords

Comments

Total number of smallest parts in all partitions of n into squarefree parts (A005117).

Examples

			a(5) = 13 because we have [5], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1] and 1 + 1 + 2 + 1 + 3 + 5 = 13.

Links

Crossrefs

Cf. A005117, A008683, A073576, A092268, A092269, A195820, A281572.

Programs

  • Mathematica
    nmax = 50; Rest[CoefficientList[Series[Sum[MoebiusMu[i]^2 x^i/(1 - x^i) Product[1/(1 - MoebiusMu[j]^2 x^j), {j, i, nmax}], {i, 1, nmax}], {x, 0, nmax}], x]]
  • PARI
    x='x+O('x^50); Vec(sum(i=1, 50, moebius(i)^2*x^i/(1 - x^i) * prod(j=i, 50, 1/(1 - moebius(j)^2*x^j)))) \\ Indranil Ghosh, Apr 04 2017

Formula

G.f.: Sum_{i>=1} mu(i)^2*x^i/(1 - x^i) * Product_{j>=i} 1/(1 - mu(j)^2*x^j).

A284830 Expansion of Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) * Product_{j>=i} 1/(1 - x^(j^2)).

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 10, 14, 16, 19, 23, 30, 33, 38, 44, 55, 60, 69, 77, 93, 102, 113, 126, 148, 162, 177, 198, 226, 246, 268, 293, 334, 361, 392, 424, 480, 516, 556, 601, 668, 721, 773, 835, 917, 990, 1054, 1129, 1239, 1325, 1415, 1508, 1649, 1757, 1875, 1990, 2157, 2303, 2441, 2595, 2796
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 03 2017

Keywords

Comments

Total number of smallest parts in all partitions of n into squares (A000290).

Examples

			a(9) = 16 because we have [9], [4, 4, 1], [4, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1] and 1 + 1 + 5 + 9 = 16.

Links

Crossrefs

Cf. A000290, A001156, A092268, A092269, A195820, A281541.

Programs

  • Mathematica
    nmax = 60; Rest[CoefficientList[Series[Sum[x^i^2/(1 - x^i^2) Product[1/(1 - x^j^2), {j, i, nmax}], {i, 1, nmax}], {x, 0, nmax}], x]]
  • PARI
    x='x+O('x^61); Vec(sum(i=1, 60, x^i^2/(1 - x^i^2) * prod(j=i, 60, 1/(1 - x^j^2)))) \\ Indranil Ghosh, Apr 04 2017

Formula

G.f.: Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) * Product_{j>=i} 1/(1 - x^(j^2)).

A284831 Expansion of Sum_{i>=1} x^(i^3)/(1 - x^(i^3)) * Product_{j>=i} 1/(1 - x^(j^3)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 16, 18, 20, 22, 26, 27, 30, 33, 36, 39, 42, 45, 51, 52, 56, 61, 65, 70, 75, 80, 89, 91, 97, 104, 110, 117, 124, 131, 143, 146, 154, 164, 171, 180, 189, 198, 213, 217, 227, 240, 248, 259, 272, 282, 301, 307, 320, 337, 347, 361, 376, 390, 414, 422, 439, 461, 474, 492, 512
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 03 2017

Keywords

Comments

Total number of smallest parts in all partitions of n into cubes (A000578).

Examples

			a(10) = 12 because we have [8, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1] and 2 + 10 = 12.

Links

Crossrefs

Cf. A000578, A003108, A092268, A092269, A195820, A281613.

Programs

  • Mathematica
    nmax = 70; Rest[CoefficientList[Series[Sum[x^i^3/(1 - x^i^3) Product[1/(1 - x^j^3), {j, i, nmax}], {i, 1, nmax}], {x, 0, nmax}], x]]

Formula

G.f.: Sum_{i>=1} x^(i^3)/(1 - x^(i^3)) * Product_{j>=i} 1/(1 - x^(j^3)).

A115604 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts in which the smallest part occurs k times (1<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 1, 0, 1, 0, 0, 1, 2, 1, 1, 0, 1, 0, 0, 1, 2, 1, 2, 1, 0, 1, 0, 0, 1, 3, 2, 1, 1, 1, 0, 1, 0, 0, 1, 3, 3, 1, 1, 1, 1, 0, 1, 0, 0, 1, 4, 2, 2, 2, 1, 1, 1, 0, 1, 0, 0, 1, 5, 3, 2, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 5, 4, 3, 2, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1
Offset: 1

Views

Author

Emeric Deutsch, Mar 13 2006

Keywords

Comments

Row sums yield A000009. T(n,1)=A087897(n+2). Sum(k*T(n,k),k=1..n)=A092268(n).

Examples

			T(14,2)=4 because we have [9,3,1,1],[7,7],[7,5,1,1] and [3,3,3,3,1,1].
Triangle starts:
1;
0,1;
1,0,1;
1,0,0,1;
1,1,0,0,1;
1,1,1,0,0,1;
2,1,0,1,0,0,1;

Crossrefs

Cf. A000009, A087897, A092268, A117408.

Programs

  • Maple
    g:=sum(t*x^(2*k-1)/(1-t*x^(2*k-1))/product(1-x^(2*i-1),i=k+1..40),k=1..40): gser:=simplify(series(g,x=0,55)): for n from 1 to 15 do P[n]:=expand(coeff(gser,x^n)) od: for n from 1 to 15 do seq(coeff(P[n],t^j),j=1..n) od; # yields sequence in triangular form

Formula

G.f.=G(t,x)=sum(tx^(2k-1)/[(1-tx^(2k-1))product(1-x^(2i-1), i=k+1..infinity)], k=1..infinity).
Previous Showing 11-15 of 15 results.

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