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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A321286 Expansion of Product_{1 <= i < j} (1 + x^(i*j)).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 5, 6, 8, 10, 14, 16, 22, 28, 33, 43, 55, 64, 84, 102, 123, 153, 188, 224, 277, 335, 401, 486, 589, 695, 843, 1006, 1191, 1428, 1698, 1999, 2384, 2814, 3312, 3914, 4612, 5395, 6355, 7447, 8691, 10182, 11892, 13826, 16146, 18770, 21779, 25313
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2018

Keywords

Links

Crossrefs

Cf. A056924, A107742, A211856, A321285.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + x^k)^Floor[DivisorSigma[0, k]/2], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 02 2018 *)
nmax = 100; A056924 = Table[Floor[DivisorSigma[0, k]/2], {k, 1, nmax}]; s = 1; Do[s *= Sum[Binomial[A056924[[k]], j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; Take[CoefficientList[s, x],nmax+1] (* Vaclav Kotesovec, Nov 02 2018 *)

Formula

G.f.: Product_{k>0} (1 + x^k)^A056924(k).

A328775 Product_{n>=1} (1 + x^n)^a(n) = 1 + Sum_{n>=1} tau(n) * x^n, where tau = A000005.

Original entry on oeis.org

1, 2, 0, 2, -1, 1, -1, 2, 1, -2, 0, 2, -1, -2, 2, 3, -2, -1, 2, 1, -4, 0, 3, -1, 3, -3, -2, 0, 1, 9, -15, 3, 17, -13, -1, -1, 9, -2, -18, 27, -10, -14, 24, -17, -15, 24, 27, -43, -37, 72, 43, -116, -11, 147, -98, -24, 67, -56, 24, -44, 213, -258, -193, 707, -435
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 27 2019

Keywords

Comments

Inverse weigh transform of A000005.

Links

Crossrefs

Cf. A000005, A107742, A320779.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; numtheory[tau](n)-b(n, n-1) end:
seq(a(n), n=1..80);  # Alois P. Heinz, Oct 27 2019
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := a[n] = DivisorSigma[0, n] - b[n, n - 1]; Array[a, 65]

A341506 E.g.f.: Product_{i>=1, j>=1} (1 + x^(i*j) / (i*j)!).

Original entry on oeis.org

1, 1, 2, 8, 17, 87, 366, 1514, 8770, 45585, 267586, 1612624, 11914416, 73215391, 522906754, 4364545708, 33150679697, 263662491935, 2151338992440, 20815916251604, 178593028936507, 1714283809331191, 15531842607259512, 158682350653110712, 1667852117293837230
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 13 2021

Keywords

Links

Crossrefs

Cf. A000005, A007837, A032315, A107742, A318694, A340903, A341505, A341876.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[Product[(1 + x^k/k!)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, -(n - 1)! Sum[Sum[d DivisorSigma[0, d]/(-d!)^(k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 24}]

Formula

E.g.f.: Product_{k>=1} (1 + x^k / k!)^sigma_0(k).

A356931 Number of multiset partitions of the prime indices of n into multisets of odd numbers. Number of factorizations of n into members of A066208.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Sep 08 2022

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The a(440) = 21 multiset partitions of {1,1,1,3,5}:
  {1}{1}{1}{3}{5}  {1}{1}{1}{35}  {1}{1}{135}  {1}{1135}  {11135}
                   {1}{1}{13}{5}  {1}{11}{35}  {11}{135}
                   {1}{11}{3}{5}  {11}{13}{5}  {111}{35}
                   {1}{1}{3}{15}  {1}{13}{15}  {113}{15}
                                  {11}{3}{15}  {13}{115}
                                  {1}{3}{115}  {3}{1115}
                                  {1}{5}{113}  {5}{1113}
                                  {3}{111}{5}

Links

Crossrefs

Positions of 0's are A324929, complement A066208.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Other types: A000009, A000012, A000041, A089259, A356930.
Other conditions: A050320, A050330, A356936, A322585, A356233, A356945.
Cf. A003963, A073491, A107742, A287170, A325534, A325535, A340852, A356234.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],And@@(OddQ[Times@@primeMS[#]]&/@#)&]],{n,100}]

Formula

a(n) = 0 if n is in A324929, otherwise a(n) = A001055(n).

A280452 G.f.: Product_{k>=1, j>=1} (1 + x^(j^2*k^2)).

Original entry on oeis.org

1, 1, 0, 0, 2, 2, 0, 0, 1, 3, 2, 0, 0, 4, 4, 0, 3, 5, 3, 1, 6, 6, 2, 2, 3, 11, 9, 1, 0, 16, 16, 0, 3, 11, 15, 7, 10, 10, 14, 14, 11, 23, 19, 9, 6, 36, 32, 4, 5, 31, 46, 18, 16, 26, 48, 36, 25, 35, 38, 36, 20, 60, 60, 28, 20, 82, 98, 30, 31, 65, 104, 64, 40
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 03 2017

Keywords

Links

Crossrefs

Cf. A107742, A280451, A280453.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + x^(j^2*k^2)), {k, 1, Floor[Sqrt[nmax]]+1}, {j, 1, Floor[Sqrt[nmax/k^2]] + 1}], {x, 0, nmax}], x]

A301746 Expansion of Product_{k>=1} (1 + x^k)^(sigma_0(k)^2).

Original entry on oeis.org

1, 1, 4, 8, 19, 35, 82, 142, 291, 524, 989, 1724, 3174, 5393, 9517, 16064, 27464, 45481, 76357, 124402, 204497, 329559, 532316, 846564, 1349481, 2120814, 3335819, 5191522, 8070062, 12434176, 19136484, 29215324, 44531151, 67431985, 101882975, 153055897
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 26 2018

Keywords

Links

Crossrefs

Cf. A000005, A035116, A107742, A301747.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1+x^k)^(DivisorSigma[0, k]^2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, k]^2, j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]]; , {k, 2, nmax}]; CoefficientList[s, x] (* Vaclav Kotesovec, Aug 29 2018 *)

Formula

Conjecture: log(a(n)) ~ sqrt(n) * log(n)^(3/2) / (2*sqrt(6)). - Vaclav Kotesovec, Aug 29 2018

A318484 Expansion of Product_{k>=1} (1 + k*x^k)^sigma(k), where sigma = A000203.

Original entry on oeis.org

1, 1, 6, 18, 52, 142, 404, 1018, 2624, 6645, 16124, 38857, 92245, 214841, 494098, 1125062, 2522188, 5604930, 12327860, 26838595, 57913194, 123951482, 263019720, 553989989, 1158449522, 2405179547, 4961047246, 10168544537, 20714279168, 41952595411, 84494479578
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 27 2018

Keywords

Links

Crossrefs

Cf. A000203, A022629, A266891, A318416, A107742, A192065, A318483.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[(1+k*x^k)^DivisorSigma[1, k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[1, k], j]*k^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[s, x]

A321877 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^sigma_k(j).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 5, 7, 6, 1, 1, 9, 15, 14, 10, 1, 1, 17, 37, 41, 28, 17, 1, 1, 33, 99, 137, 107, 58, 25, 1, 1, 65, 277, 491, 487, 286, 106, 38, 1, 1, 129, 795, 1829, 2429, 1749, 700, 201, 59, 1, 1, 257, 2317, 6971, 12763, 12056, 5901, 1735, 372, 86
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 20 2018

Keywords

Examples

			Square array begins:
   1,   1,    1,    1,     1,      1,  ...
   1,   1,    1,    1,     1,      1,  ...
   2,   3,    5,    9,    17,     33,  ...
   4,   7,   15,   37,    99,    277,  ...
   6,  14,   41,  137,   491,   1829,  ...
  10,  28,  107,  487,  2429,  12763,  ...

Links

Crossrefs

Columns k=0..9 give A107742, A192065, A288414, A288415, A301548, A301549, A301550, A301551, A301552, A301553.
Main diagonal gives A321042.
Cf. A321876.

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[Product[(1 + x^j)^DivisorSigma[k, j], {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 10}, {n, 0, i}] // Flatten
Table[Function[k, SeriesCoefficient[Exp[Sum[DivisorSigma[k + 1, j] x^j/(j (1 - x^(2 j))), {j, 1, n}]], {x, 0, n}]][i - n], {i, 0, 10}, {n, 0, i}] // Flatten

Formula

G.f. of column k: Product_{i>=1, j>=1} (1 + x^(i*j))^(j^k).
G.f. of column k: exp(Sum_{j>=1} sigma_(k+1)(j)*x^j/(j*(1 - x^(2*j)))).

A327731 Expansion of Product_{i>=1, j>=1} (1 + x^(i*(2*j - 1))).

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 8, 10, 14, 21, 28, 36, 51, 65, 86, 117, 148, 190, 251, 316, 402, 519, 647, 814, 1032, 1282, 1593, 1994, 2457, 3029, 3754, 4591, 5617, 6895, 8381, 10193, 12411, 14999, 18125, 21919, 26359, 31672, 38074, 45556, 54468, 65134, 77576, 92322
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 23 2019

Keywords

Comments

Weigh transform of A001227.

Crossrefs

Cf. A001227, A107742, A301800, A320650.

Programs

  • Mathematica
    nmax = 47; CoefficientList[Series[Product[(1 + x^k)^DivisorSum[k, Mod[#, 2] &], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d DivisorSum[d, Mod[#, 2] &], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 47}]
  • PARI
    seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^numdiv(k>>valuation(k, 2))))} \\ Andrew Howroyd, Sep 23 2019

Formula

G.f.: Product_{k>=1} (1 + x^k)^A001227(k).

A327851 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A111374.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 8, 12, 15, 19, 24, 30, 36, 47, 57, 74, 88, 112, 130, 160, 190, 232, 277, 333, 399, 471, 554, 656, 768, 908, 1060, 1256, 1452, 1702, 1968, 2294, 2646, 3068, 3549, 4093, 4710, 5418, 6211, 7121, 8138, 9331, 10625, 12150, 13817, 15749, 17858, 20290, 23000, 26054
Offset: 0

Views

Author

Seiichi Manyama, Sep 28 2019

Keywords

Comments

a(n) > 0.

Links

Crossrefs

Convolution inverse of A327852.
Product_{k>=1} (1 - x^k)^(- Sum_{d|k} (b/d)), where (m/n) is the Kronecker symbol: this sequence (b=2), A107742 (b=4), A327716 (b=5).
Cf. A035185, A111374.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Product[QPochhammer[x^(8*j - 3)] * QPochhammer[x^(8*j - 5)]/(QPochhammer[x^(8*j - 7)] * QPochhammer[x^(8*j - 1)]), {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 28 2019 *)
  • PARI
    N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1-x^k)^sumdiv(k, d, kronecker(2, d))))

Formula

G.f.: Product_{i>=1} Product_{j>=1} (1-x^(i*(8*j-3))) * (1-x^(i*(8*j-5))) / ((1-x^(i*(8*j-1))) * (1-x^(i*(8*j-7)))).
G.f.: Product_{k>=1} (1-x^k)^(-A035185(k)).
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