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.

Previous Showing 41-47 of 47 results.

A117586 Coefficients of q in series expansion of Zagier's identity.

Original entry on oeis.org

0, -1, -2, -1, -1, 2, 0, 4, 1, 2, 1, 2, -4, 1, -1, -5, -2, -1, -3, -1, -2, -2, 5, 0, -1, 1, 8, 0, 3, 2, 2, 2, 3, 0, 4, -7, 0, 0, 2, -3, -8, -2, -1, -3, -2, -4, 0, -3, -3, -2, -1, 7, -1, 0, 1, -1, 0, 12, 2, 2, 0, 4, 3, 4, 0, 2, 4, 3, 0, 5, -12, 2, 0, 1, -1, 1, -3, -11, -1, -2, -6, 2, -4, -3, -3, -4, -2, 1, -5, -3, -3, -2, 11, 2, -2, -3, 2, 0, 0, 3, 12, 1
Offset: 0

Views

Author

Eric W. Weisstein, Mar 29 2006

Keywords

Examples

			G.f. = - x - 2*x^2 - x^3 - x^4 + 2*x^5 + 4*x^7 + x^8 + 2*x^9 + x^10 + ...

Links

Crossrefs

Cf. A046746.

Programs

  • Mathematica
    Flatten[{0, CoefficientList[Series[-Sum[x^(n - 1)*(QPochhammer[x^(n + 1), x]^2/QPochhammer[x^(n), x]), {n, 1, 101}], {x, 0, 100}], x]}] (* Mats Granvik, Jan 05 2015 *)
a[ n_] := SeriesCoefficient[ Sum[ QPochhammer[ x] - QPochhammer[ x, x, k], {k, 0, n}], {x, 0, n}]; (* Michael Somos, Jan 07 2015 *)
a[ n_] := SeriesCoefficient[ -Sum[ QPochhammer[ x^k, x] x^k / (1 - x^k)^2, {k, n}], {x, 0, n}]; (* Michael Somos, Jan 07 2015 *)

Formula

Negative of sequence is convolution of A010815 with A046746. - Michael Somos, Jan 07 2015
a(n) = A067661(n) - A067659(n) [Chapman]. - George Beck, May 06 2017

A210946 Triangle read by rows: T(n,k) = sum of parts in the k-th column of the mirror of the last section of the set of partitions of n with its parts aligned to the right margin.

Original entry on oeis.org

1, 3, 5, 9, 2, 12, 3, 20, 9, 2, 25, 11, 3, 38, 22, 9, 2, 49, 28, 14, 3, 69, 44, 26, 9, 2, 87, 55, 37, 14, 3, 123, 83, 62, 29, 9, 2, 152
Offset: 1

Views

Author

Omar E. Pol, Apr 21 2012

Keywords

Comments

Row n lists the positive terms of the n-th row of triangle A210953 in decreasing order.

Examples

			For n = 7 the illustration shows two arrangements of the last section of the set of partitions of 7:
.
.       (7)        (7)
.     (4+3)        (3+4)
.     (5+2)        (2+5)
.   (3+2+2)        (2+2+3)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.                 ---------
.                  25,11,3
.
The left hand picture shows the last section of 7 with its parts aligned to the right margin. In the right hand picture (the mirror) we can see that the sum of all parts of the columns 1..3 are 25, 11, 3 therefore row 7 lists 25, 11, 3.
Written as a triangle begins:
1;
3;
5;
9,    2;
12,   3;
20,   9,  2;
25,  11,  3;
38,  22,  9,  2;
49,  28, 14,  3;
69,  44, 26,  9,  2;
87,  55, 37, 14,  3,
123, 83, 62, 29,  9,  2;

Crossrefs

Column 1 is A046746, n >= 1. Row sums give A138879.
Cf. A135010, A138121, A182703, A194714, A196807, A206437, A207031, A207034, A207035, A210945, A210952, A210953.

A284833 Expansion of Sum_{i>=1} x^prime(i)/(1 - x^prime(i)) * Product_{j=1..i} 1/(1 - x^prime(j)).

Original entry on oeis.org

0, 1, 1, 2, 2, 5, 3, 7, 6, 11, 8, 17, 12, 22, 21, 28, 27, 41, 35, 53, 52, 66, 66, 90, 85, 112, 114, 140, 143, 182, 180, 219, 236, 269, 291, 342, 353, 417, 444, 508, 540, 625, 657, 751, 812, 901, 974, 1097, 1168, 1313, 1414, 1562, 1684, 1874, 2008, 2219, 2397, 2626, 2832, 3121, 3341, 3668, 3956, 4305, 4650
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 03 2017

Keywords

Comments

Total number of largest parts in all partitions of n into prime parts.

Examples

			a(10) = 11 because we have [7, 3], [5, 5], [5, 3, 2], [3, 3, 2, 2], [2, 2, 2, 2, 2] and 1 + 2 + 1 + 2 + 5 = 11.

Links

Crossrefs

Cf. A000607, A046746, A084993, A092311, A281544, A284827.

Programs

  • Mathematica
    nmax = 65; Rest[CoefficientList[Series[Sum[x^Prime[i]/(1 - x^Prime[i]) Product[1/(1 - x^Prime[j]), {j, 1, i}], {i, 1, nmax}], {x, 0, nmax}], x]]
  • PARI
    x='x+O('x^66); concat([0], Vec(sum(i=1, 66, x^prime(i)/(1 - x^prime(i)) * prod(j=1,i, 1/(1 - x^prime(j)))))) \\ Indranil Ghosh, Apr 04 2017

Formula

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

A284834 Expansion of Sum_{i>=2} x^prime(i)/(1 - x^prime(i)) * Product_{j=2..i} 1/(1 - x^prime(j)).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 1, 3, 3, 2, 5, 4, 4, 9, 5, 6, 12, 8, 11, 17, 12, 14, 23, 19, 21, 29, 27, 29, 41, 37, 36, 56, 49, 55, 72, 62, 74, 91, 90, 96, 116, 117, 125, 155, 149, 162, 195, 194, 215, 246, 248, 270, 311, 324, 344, 389, 406, 435, 494, 509, 546, 615, 636, 694, 763, 787, 861, 942, 994, 1063
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 03 2017

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A046746, A065091, A084993, A092311, A099773, A284828.

Programs

  • Mathematica
    nmax = 64; Rest[CoefficientList[Series[Sum[x^Prime[i]/(1 - x^Prime[i]) Product[1/(1 - x^Prime[j]), {j, 2, i}], {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=2,i, 1/(1 - x^prime(j)))))) \\ Indranil Ghosh, Apr 04 2017

Formula

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

A284835 Expansion of Sum_{i>=1} mu(i)^2*x^i/(1 - x^i) * Product_{j=1..i} 1/(1 - mu(j)^2*x^j), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 3, 5, 8, 11, 18, 22, 31, 39, 53, 64, 87, 104, 134, 165, 205, 248, 310, 368, 455, 545, 659, 784, 947, 1117, 1337, 1579, 1872, 2197, 2604, 3036, 3570, 4168, 4866, 5661, 6599, 7633, 8859, 10236, 11831, 13625, 15715, 18036, 20728, 23761, 27211, 31106, 35560, 40533, 46221, 52596, 59813, 67912, 77090, 87343
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 03 2017

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A005117, A008683, A046746, A073576, A092311, A281572, A284829.

Programs

  • Mathematica
    nmax = 55; Rest[CoefficientList[Series[Sum[MoebiusMu[i]^2 x^i/(1 - x^i) Product[1/(1 - MoebiusMu[j]^2 x^j), {j, 1, i}], {i, 1, nmax}], {x, 0, nmax}], x]]
  • PARI
    x='x+O('x^56); Vec(sum(i=1, 56, moebius(i)^2*x^i/(1 - x^i) * prod(j=1, i, 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=1..i} 1/(1 - mu(j)^2*x^j).

A284836 Expansion of Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) * Product_{j=1..i} 1/(1 - x^(j^2)).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 11, 13, 14, 15, 19, 21, 22, 23, 29, 31, 34, 35, 42, 44, 47, 48, 56, 60, 63, 67, 76, 80, 83, 87, 99, 103, 108, 112, 130, 134, 139, 143, 162, 169, 174, 180, 200, 213, 218, 224, 248, 262, 272, 278, 306, 320, 337, 343, 372, 390, 408, 419, 449, 471, 489, 508, 544, 567, 591, 611, 654, 677, 705
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 03 2017

Keywords

Comments

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

Examples

			a(9) = 13 because we have [9], [4, 4, 1], [4, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1] and 1 + 2 + 1 + 9 = 13.

Links

Crossrefs

Cf. A000290, A001156, A046746, A092311, A281541, A284830.

Programs

  • Mathematica
    nmax = 70; Rest[CoefficientList[Series[Sum[x^i^2/(1 - x^i^2) Product[1/(1 - x^j^2), {j, 1, i}], {i, 1, nmax}], {x, 0, nmax}], x]]
  • PARI
    x='x+O('x^71); Vec(sum(i=1, 71, x^(i^2)/(1 - x^(i^2)) * prod(j=1, i, 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=1..i} 1/(1 - x^(j^2)).

A284837 Expansion of Sum_{i>=1} x^(i^3)/(1 - x^(i^3)) * Product_{j=1..i} 1/(1 - x^(j^3)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 24, 25, 26, 30, 31, 32, 34, 35, 36, 37, 38, 43, 44, 45, 47, 48, 49, 50, 51, 57, 58, 59, 61, 62, 63, 64, 65, 72, 73, 74, 76, 77, 78, 81, 82, 90, 91, 92, 94, 95, 96, 99, 100, 110, 111, 112, 114, 115, 116, 119, 120, 131, 132, 133, 135
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 03 2017

Keywords

Comments

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

Examples

			a(10) = 11 because we have [8, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1] and 1 + 10 = 11.

Links

Crossrefs

Cf. A000578, A003108, A046746, A092311, A281613, A284831.

Programs

  • Mathematica
    nmax = 75; Rest[CoefficientList[Series[Sum[x^i^3/(1 - x^i^3) Product[1/(1 - x^j^3), {j, 1, i}], {i, 1, nmax}], {x, 0, nmax}], x]]

Formula

G.f.: Sum_{i>=1} x^(i^3)/(1 - x^(i^3)) * Product_{j=1..i} 1/(1 - x^(j^3)).
Previous Showing 41-47 of 47 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.