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 1-10 of 25 results. Next

A032302 G.f.: Product_{k>=1} (1 + 2*x^k).

Original entry on oeis.org

1, 2, 2, 6, 6, 10, 18, 22, 30, 42, 66, 78, 110, 138, 186, 254, 318, 402, 522, 654, 822, 1074, 1306, 1638, 2022, 2514, 3058, 3798, 4662, 5658, 6882, 8358, 10062, 12186, 14610, 17534, 21150, 25146, 29994, 35694, 42446, 50178, 59514, 70110, 82758, 97602, 114570, 134262
Offset: 0

Views

Author

Christian G. Bower, Apr 01 1998

Keywords

Comments

"EFK" (unordered, size, unlabeled) transform of 2,2,2,2,...
Number of partitions into distinct parts of 2 sorts, see example. - Joerg Arndt, May 22 2013
In general, for a fixed integer m > 0, if g.f. = Product_{k>=1} (1 + m*x^k) then a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt((m+1)*Pi)*n^(3/4)), where c = Pi^2/6 + log(m)^2/2 + polylog(2, -1/m) = -polylog(2, -m). - Vaclav Kotesovec, Jan 04 2016
Antidiagonal sums of A284593. - Peter Bala, Mar 30 2017

Examples

			From _Joerg Arndt_, May 22 2013: (Start)
There are a(7) = 22 partitions of 7 into distinct parts of 2 sorts (format P:S for part:sort):
01:  [ 1:0  2:0  4:0  ]
02:  [ 1:0  2:0  4:1  ]
03:  [ 1:0  2:1  4:0  ]
04:  [ 1:0  2:1  4:1  ]
05:  [ 1:0  6:0  ]
06:  [ 1:0  6:1  ]
07:  [ 1:1  2:0  4:0  ]
08:  [ 1:1  2:0  4:1  ]
09:  [ 1:1  2:1  4:0  ]
10:  [ 1:1  2:1  4:1  ]
11:  [ 1:1  6:0  ]
12:  [ 1:1  6:1  ]
13:  [ 2:0  5:0  ]
14:  [ 2:0  5:1  ]
15:  [ 2:1  5:0  ]
16:  [ 2:1  5:1  ]
17:  [ 3:0  4:0  ]
18:  [ 3:0  4:1  ]
19:  [ 3:1  4:0  ]
20:  [ 3:1  4:1  ]
21:  [ 7:0  ]
22:  [ 7:1  ]
(End)

Links

Crossrefs

Cf. A000009, A032308, A261562, A261568, A261569, A266576, A284593.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, 2*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 24 2015
# Alternatively:
simplify(expand(QDifferenceEquations:-QPochhammer(-2,x,99)/3,x)):
seq(coeff(%,x,n), n=0..47); # Peter Luschny, Nov 17 2016
  • Mathematica
    nn=47; CoefficientList[Series[Product[1+2x^i,{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Sep 07 2013 *)
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*2^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(QPochhammer[-2, x]/3 + O[x]^58)[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(n=1,N, 1+2*x^n)) \\ Joerg Arndt, May 22 2013

Formula

a(n) = A072706(n)*2 for n>=1.
G.f.: Sum_{n>=0} (2^n*q^(n*(n+1)/2) / Product_{k=1..n} (1-q^k ) ). - Joerg Arndt, Jan 20 2014
a(n) = (1/3) [x^n] QPochhammer(-2,x). - Vladimir Reshetnikov, Nov 20 2015
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(3*Pi)*n^(3/4)), where c = Pi^2/6 + log(2)^2/2 + polylog(2, -1/2) = 1.43674636688368094636290202389358335424... . Equivalently, c = A266576 = Pi^2/12 + log(2)^2 + polylog(2, 1/4)/2. - Vaclav Kotesovec, Jan 04 2016

A344062 Expansion of Product_{k>=1} (1 + 3^(k-1)*x^k).

Original entry on oeis.org

1, 1, 3, 12, 36, 135, 432, 1539, 4860, 17496, 55404, 192456, 623295, 2125764, 6849684, 23442453, 75110328, 252965916, 822670668, 2735858268, 8838926712, 29501352792, 95090206689, 314068876416, 1018141045092, 3342663979092, 10798571289897, 35481518064576
Offset: 0

Views

Author

Ilya Gutkovskiy, May 08 2021

Keywords

Crossrefs

Cf. A003056, A008289, A032308, A300579, A304961, A340103, A344063, A344064, A344065, A344066, A344067, A344068.

Programs

  • Mathematica
    nmax = 27; CoefficientList[Series[Product[(1 + 3^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 3^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 27}]
  • PARI
    seq(n)={Vec(prod(k=1, n, 1 + 3^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

Formula

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 3^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.
a(n) ~ (-polylog(2, -1/3))^(1/4) * 3^n * exp(2*sqrt(-polylog(2, -1/3)*n)) / (4*sqrt(Pi/3)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A261568 Expansion of Product_{k>=1} (1 + 4*x^k).

Original entry on oeis.org

1, 4, 4, 20, 20, 36, 100, 116, 180, 260, 580, 660, 1044, 1380, 2020, 3444, 4340, 6020, 8260, 11220, 14740, 23140, 28196, 38900, 50420, 67780, 85956, 114900, 157140, 197860, 257060, 331060, 423540, 540100, 687620, 864084, 1145300, 1406500, 1789860, 2231860
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 24 2015

Keywords

Links

Crossrefs

Cf. A000009, A032302, A032308, A261569.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, 4*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 24 2015
  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1 + 4*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*4^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(QPochhammer[-4, x]/5 + O[x]^58)[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

Formula

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(5*Pi)*n^(3/4)), where c = Pi^2/6 + 2*log(2)^2 + polylog(2, -1/4) = 2.36993979699836583198553742535032304875... . - Vaclav Kotesovec, Jan 04 2016
G.f.: Sum_{i>=0} 4^i*x^(i*(i+1)/2)/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018

A261582 Expansion of Product_{k>=1} 1/(1 + 3*x^k).

Original entry on oeis.org

1, -3, 6, -21, 69, -201, 591, -1785, 5406, -16194, 48426, -145380, 436641, -1309611, 3927399, -11783280, 35354139, -106059387, 318165729, -954506190, 2863556475, -8590643832, 25771817454, -77315531169, 231946940175, -695840583126, 2087520715788, -6262562872614
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 25 2015

Keywords

Links

Crossrefs

Cf. A032308, A242587, A246935, A261565, A261567.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1 + 3*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^k*3^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x]
(O[x]^30 + 4/QPochhammer[-3, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

Formula

a(n) ~ c * (-3)^n, where c = Product_{j>=1} 1/(1-1/(-3)^j) = 1/QPochhammer[-1/3,-1/3] = 0.8212554466473167689981660621182786378...
G.f.: Sum_{i>=0} (-3)^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 13 2018

A261569 Expansion of Product_{k>=1} (1 + 5*x^k).

Original entry on oeis.org

1, 5, 5, 30, 30, 55, 180, 205, 330, 480, 1230, 1380, 2255, 3030, 4530, 8555, 10680, 15330, 21330, 29730, 39480, 67380, 81505, 116280, 153030, 210930, 270805, 370080, 534330, 675480, 900480, 1180380, 1544130, 1997280, 2597280, 3304805, 4581180, 5653080
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 24 2015

Keywords

Comments

In general, for a fixed integer m > 0, if g.f. = Product_{k>=1} (1 + m*x^k) then a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt((m+1)*Pi)*n^(3/4)), where c = Pi^2/6 + log(m)^2/2 + polylog(2, -1/m) = -polylog(2, -m). - Vaclav Kotesovec, Jan 04 2016

Links

Crossrefs

Cf. A000009, A032302, A032308, A261568, A291698.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, 5*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 24 2015
  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1 + 5*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*5^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(QPochhammer[-5, x]/6 + O[x]^58)[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

Formula

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(6*Pi)*n^(3/4)), where c = Pi^2/6 + log(5)^2/2 + polylog(2, -1/5) = 2.74927912606080829002558751537626864449... . - Vaclav Kotesovec, Jan 04 2016
G.f.: Sum_{i>=0} 5^i*x^(i*(i+1)/2)/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018

A286957 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + k*x^j).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 2, 0, 1, 4, 3, 6, 2, 0, 1, 5, 4, 12, 6, 3, 0, 1, 6, 5, 20, 12, 10, 4, 0, 1, 7, 6, 30, 20, 21, 18, 5, 0, 1, 8, 7, 42, 30, 36, 48, 22, 6, 0, 1, 9, 8, 56, 42, 55, 100, 57, 30, 8, 0, 1, 10, 9, 72, 56, 78, 180, 116, 84, 42, 10, 0, 1, 11, 10, 90, 72, 105, 294, 205, 180, 120, 66, 12, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, May 17 2017

Keywords

Comments

A(n,k) is the number of partitions of n into distinct parts of k sorts: the parts are unordered, but not the sorts.

Examples

			Square array begins:
1,  1,   1,   1,   1,   1,  ...
0,  1,   2,   3,   4,   5,  ...
0,  1,   2,   3,   4,   5,  ...
0,  2,   6,  12,  20,  30,  ...
0,  2,   6,  12,  20,  30,  ...
0,  3,  10,  21,  36,  55,  ...

Links

Crossrefs

Columns k=0-5 give: A000007, A000009, A032302, A032308, A261568, A261569.
Main diagonal gives A291698.
Cf. A246935.

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[Product[(1 + k x^i), {i, 1, Infinity}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[-k, x]/(1 + k), {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

Formula

G.f. of column k: Product_{j>=1} (1 + k*x^j).

A292128 Expansion of Product_{k>=1} (1 - 3*x^k).

Original entry on oeis.org

1, -3, -3, 6, 6, 15, -12, -3, -30, -48, 6, -12, 15, 78, 186, -21, 168, 42, 42, -246, -408, -156, -399, -552, -498, -246, 213, 1248, -318, 1608, 1392, 2508, 1482, 2976, -480, -1011, 1500, -1704, -4296, -4206, -8499, -8652, -7626, -7050, -192, -13008, -480, -2118
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2017

Keywords

Links

Crossrefs

Column k=3 of A292131.
Product_{k>=1} (1 - m*x^k): A010815 (m=1), A070877 (m=2), this sequence (m=3).
Cf. A032308, A242587, A292130.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(prod(n=1, N, 1-3*x^n))

Formula

G.f.: Sum_{i>=0} (-3)^i*x^(i*(i+1)/2)/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 13 2018

A266821 Expansion of Product_{k>=1} (1 + 3*x^k) / (1 - x^k).

Original entry on oeis.org

1, 4, 8, 24, 44, 88, 176, 312, 544, 924, 1584, 2552, 4136, 6488, 10128, 15632, 23748, 35640, 53080, 78136, 114024, 165552, 237744, 339544, 481248, 678236, 949008, 1321840, 1830376, 2521688, 3456672, 4717208, 6406680, 8666448, 11672464, 15660528, 20934868
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 04 2016

Keywords

Comments

Convolution of A000041 and A032308.
In general, for m > 0, if g.f. = Product_{k>=1} ((1 + m*x^k) / (1 - x^k)) then a(n) ~ sqrt(c) * exp(sqrt(2*c*n)) / (4*Pi*sqrt(m+1)*n), where c = 2*Pi^2/3 + log(m)^2 + 2*polylog(2, -1/m).

Links

Crossrefs

Cf. A000041, A015128, A032308, A264686.
Column k=4 of A321884.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      (t-> b(t, min(t, i-1)))(n-i*j), j=1..n/i)*4 +b(n, i-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..44);  # Alois P. Heinz, Aug 28 2019
  • Mathematica
    nmax = 40; CoefficientList[Series[Product[(1+3*x^k) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]
  • PARI
    { my(n=40); Vec(prod(k=1, n, 4/(1-x^k) - 3 + O(x*x^n))) } \\ Andrew Howroyd, Dec 22 2017

Formula

a(n) ~ sqrt(c) * exp(sqrt(2*c*n)) / (8*Pi*n), where c = 2*Pi^2/3 + log(3)^2 + 2*polylog(2, -1/3) = 7.16861897522987077909937377164783326088308015803... .

A279368 Expansion of Product_{k>=1} (1+3*x^(k^2)).

Original entry on oeis.org

1, 3, 0, 0, 3, 9, 0, 0, 0, 3, 9, 0, 0, 9, 27, 0, 3, 9, 0, 0, 9, 27, 0, 0, 0, 12, 36, 0, 0, 36, 108, 0, 0, 0, 9, 27, 3, 9, 27, 81, 9, 36, 27, 0, 0, 36, 108, 0, 0, 30, 117, 81, 9, 36, 108, 243, 27, 81, 9, 27, 0, 36, 135, 81, 3, 126, 351, 0, 9, 54, 108, 81, 0
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 10 2016

Keywords

Comments

In general, if m > 0 and g.f. = Product_{k>=1} (1 + m*x^(k^2)), then a(n) ~ exp(3 * 2^(-4/3) * Pi^(1/3) * c^(2/3) * n^(1/3)) * c^(1/3) / (2^(2/3) * Pi^(1/3) * sqrt(3*(m+1)) * n^(5/6)), where c = -PolyLog(3/2, -m). - Vaclav Kotesovec, Dec 12 2016

Links

Crossrefs

Cf. A032308, A033461, A279360.

Programs

  • Mathematica
    nmax = 200; CoefficientList[Series[Product[(1+3*x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 200; nn = Floor[Sqrt[nmax]]+1; poly = ConstantArray[0, nn^2 + 1]; poly[[1]] = 1; poly[[2]] = 3; poly[[3]] = 0; Do[Do[poly[[j + 1]] += 3*poly[[j - k^2 + 1]], {j, nn^2, k^2, -1}];, {k, 2, nn}]; Take[poly, nmax+1]

Formula

a(n) ~ c^(1/3) * exp(3 * 2^(-4/3) * c^(2/3) * Pi^(1/3) * n^(1/3)) / (2^(5/3) * sqrt(3) * Pi^(1/3) * n^(5/6)), where c = -PolyLog(3/2, -3) = 1.679089730504828... . - Vaclav Kotesovec, Dec 12 2016

A303390 Expansion of Product_{k>=1} (1 + 3*x^k)/(1 - 3*x^k).

Original entry on oeis.org

1, 6, 24, 96, 330, 1104, 3552, 11184, 34584, 105990, 322224, 975264, 2942016, 8857680, 26631312, 80005632, 240219114, 721036320, 2163789816, 6492625152, 19480105392, 58444390176, 175340344416, 526034008752, 1578124753152, 4734415061142, 14203316252400
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 23 2018

Keywords

Crossrefs

Cf. A032308, A242587.
Cf. A015128, A261584, A303391.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1+3*x^k)/(1-3*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ c * 3^n, where c = QPochhammer[-1, 1/3] / QPochhammer[1/3] = 5.5877920355220979147599292926505407983327527...
Showing 1-10 of 25 results. Next

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