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 11 results. Next

A321042 a(n) = [x^n] Product_{k>=1} (1 + x^k)^sigma_n(k).

Original entry on oeis.org

1, 1, 5, 37, 491, 12763, 690756, 70250881, 13805853214, 5567873958982, 4386114219458332, 6711687353310594027, 21048327399504558833175, 131214860796100022696745520, 1603892616451767287785208156624, 40296605442098101265893075903063822, 2031406440758379976992019043333960734724
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 26 2018

Keywords

Links

Crossrefs

Cf. A107742, A192065, A288414, A288415, A301548, A301549, A301550, A301551, A301552, A301553, A319647.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 + x^k)^DivisorSigma[n, k], {k, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Product[Product[(1 + x^(i j))^(j^n), {j, 1, n}], {i, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Exp[Sum[DivisorSigma[n + 1, k] x^k/(k (1 - x^(2 k))), {k, 1, n}]], {x, 0, n}], {n, 0, 16}]

Formula

a(n) = [x^n] Product_{i>=1, j>=1} (1 + x^(i*j))^(j^n).
a(n) = [x^n] exp(Sum_{k>=1} sigma_(n+1)(k)*x^k/(k*(1 - x^(2*k)))).

A321585 Number of connected nonnegative integer matrices with sum of entries equal to n and no zero rows or columns.

Original entry on oeis.org

1, 1, 3, 11, 52, 312, 2290, 19920, 200522, 2293677, 29389005, 416998371, 6490825772, 109972169413, 2014696874717, 39684502845893, 836348775861331, 18777970539419957, 447471215460930665, 11279275874429302811, 299844572529989373703, 8383794111721619471384, 245956060268568277412668
Offset: 0

Views

Author

Gus Wiseman, Nov 13 2018

Keywords

Comments

A matrix is connected if the positions in each row (or each column) of the nonzero entries form a connected hypergraph.

Examples

			The a(3) = 11 matrices:
  [3] [2 1] [1 2] [1 1 1]
.
  [2] [1 1] [1 1] [1] [1 0] [0 1]
  [1] [1 0] [0 1] [2] [1 1] [1 1]
.
  [1]
  [1]
  [1]

Links

Crossrefs

Cf. A007718, A056156, A120733, A319557, A319565, A319647, A319616-A319629, A321584.

Programs

  • Mathematica
    multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],Length[csm[Map[Last,GatherBy[#,First],{2}]]]==1]&]],{n,5}] (* Mathematica 7.0+ *)
  • PARI
    NonZeroCols(M)={my(C=Vec(M)); Mat(vector(#C, n,  sum(k=1, n, (-1)^(n-k)*binomial(n,k)*C[k])))}
ConnectedMats(M)={my([m,n]=matsize(M), R=matrix(m,n)); for(m=1, m, for(n=1, n, R[m,n] = M[m,n] - sum(i=1, m-1, sum(j=1, n-1, binomial(m-1,i-1)*binomial(n,j)*R[i,j]*M[m-i,n-j])))); R}
seq(n)={my(M=matrix(n,n,i,j,sum(k=1, n, binomial(i*j+k-1,k)*x^k, O(x*x^n) ))); Vec(1 + vecsum(vecsum(Vec( ConnectedMats( NonZeroCols( NonZeroCols(M)~))))))} \\ Andrew Howroyd, Jan 17 2024

Extensions

a(7) onwards from Andrew Howroyd, Jan 17 2024

A320940 a(n) = Sum_{d|n} d*sigma_n(d).

Original entry on oeis.org

1, 11, 85, 1127, 15631, 287021, 5764809, 135007759, 3487020610, 100146496681, 3138428376733, 107032667155169, 3937376385699303, 155582338242604221, 6568408966322733475, 295154660699054931999, 14063084452067724991027, 708239400347943609329270
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 28 2018

Keywords

Examples

			a(6) = 1*sigma_6(1)+2*sigma_6(2)+3*sigma_6(3)+6*sigma_6(6) = 1+2*65+3*730+6*47450 = 287021.

Links

Crossrefs

Cf. A000203, A001001, A027847, A027848, A319647, A321141.

Programs

  • Magma
    [&+[d*DivisorSigma(n,d):d in Divisors(n)]:n in [1..18]]; // Marius A. Burtea, Feb 15 2020
  • Maple
    with(numtheory): seq(coeff(series(n*(-log(mul((1-x^k)^sigma[n](k),k=1..n))),x,n+1), x, n), n = 1 .. 20); # Muniru A Asiru, Oct 28 2018
  • Mathematica
    Table[Sum[d DivisorSigma[n, d], {d, Divisors[n]}] , {n, 18}]
Table[n SeriesCoefficient[-Log[Product[(1 - x^k)^DivisorSigma[n, k], {k, 1, n}]], {x, 0, n}], {n, 18}]
  • PARI
    a(n) = sumdiv(n, d, d*sigma(d, n)); \\ Michel Marcus, Oct 28 2018
  • Python
    from sympy import divisor_sigma, divisors
def A320940(n):
    return sum(divisor_sigma(d)*(n//d)**(n+1) for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

Formula

a(n) = n * [x^n] -log(Product_{k>=1} (1 - x^k)^sigma_n(k)).
a(n) = Sum_{d|n} d^(n+1)*sigma_1(n/d).
a(n) ~ n^(n+1). - Vaclav Kotesovec, Feb 16 2020

A321588 Number of connected nonnegative integer matrices with sum of entries equal to n, no zero rows or columns, and distinct rows and columns.

Original entry on oeis.org

1, 1, 1, 9, 29, 181, 1285, 10635, 102355, 1118021, 13637175, 184238115, 2727293893, 43920009785, 764389610843, 14297306352937, 286014489487815, 6093615729757841, 137750602009548533, 3293082026520294529, 83006675263513350581, 2200216851785981586729, 61180266502369886181253
Offset: 0

Views

Author

Gus Wiseman, Nov 13 2018

Keywords

Comments

A matrix is connected if the positions in each row (or each column) of the nonzero entries form a connected hypergraph.

Examples

			The a(4) = 29 matrices:
4 31 13
.
3 21 21 20 12 12 11 110 11 110 101 101 1 10 10 02 011 011 01 01
1 10 01 11 10 01 20 101 02 011 110 011 3 21 12 11 110 101 21 12
.
11 11 10 10 01 01
10 01 11 01 11 10
01 10 01 11 10 11

Links

Crossrefs

Cf. A007718, A056156, A059201, A120733, A283877, A316980, A319557, A319558, A319559, A319565, A319647, A319616-A319629, A321446, A321515, A369285.

Programs

  • Mathematica
    prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@prs2mat[#],UnsameQ@@Transpose[prs2mat[#]],Length[csm[Map[Last,GatherBy[#,First],{2}]]]==1]&]],{n,6}]
  • PARI
    permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q,t,wf)={prod(j=1, #q, wf(t*q[j]))-1}
Q(m,n,wf=w->2)={my(s=0); forpart(p=m, s+=(-1)^#p*permcount(p)*exp(-sum(t=1, n, (-1)^t*x^t*K(p,t,wf)/t, O(x*x^n))) ); Vec((-1)^m*serchop(serlaplace(s),1), -n)}
ConnectedMats(M)={my([m, n]=matsize(M), R=matrix(m, n)); for(m=1, m, for(n=1, n, R[m, n] = M[m, n] - sum(i=1, m-1, sum(j=1, n-1, binomial(m-1, i-1)*binomial(n, j)*R[i, j]*M[m-i, n-j])))); R}
seq(n)={my(R=vectorv(n,m,Q(m,n,w->1/(1 - y^w) + O(y*y^n)))); for(i=2, #R, R[i] -= i*R[i-1]); Vec(1 + vecsum( vecsum( Vec( ConnectedMats( Mat(R))))))} \\ Andrew Howroyd, Jan 24 2024

Extensions

a(7) onwards from Andrew Howroyd, Jan 24 2024

A321876 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - x^j)^sigma_k(j).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 6, 8, 11, 1, 1, 10, 16, 21, 17, 1, 1, 18, 38, 52, 39, 34, 1, 1, 34, 100, 156, 128, 92, 52, 1, 1, 66, 278, 526, 534, 373, 170, 94, 1, 1, 130, 796, 1896, 2546, 2014, 913, 360, 145, 1, 1, 258, 2318, 7102, 13074, 12953, 6796, 2399, 667, 244
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,  ...
   3,   4,    6,   10,    18,     34,  ...
   5,   8,   16,   38,   100,    278,  ...
  11,  21,   52,  156,   526,   1896,  ...
  17,  39,  128,  534,  2546,  13074,  ...

Links

Crossrefs

Columns k=0..9 give A006171, A061256, A275585, A288391, A301542, A301543, A301544, A301545, A301546, A301547.
Main diagonal gives A319647.
Cf. A321877.

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[Product[1/(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^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/(1 - x^(i*j))^(j^k).
G.f. of column k: exp(Sum_{j>=1} sigma_(k+1)(j)*x^j/(j*(1 - x^j))).

A321057 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 - x^k))^sigma_n(k).

Original entry on oeis.org

1, 2, 12, 94, 1522, 48154, 3087600, 377880794, 93356591804, 46415548879976, 44773963087975388, 86770399797767582434, 340765670578000502365102, 2625605734866823121935402410, 40755373130582885082115865730892, 1290109927277547765958474680645604818
Offset: 0

Views

Author

Seiichi Manyama, Oct 26 2018

Keywords

Links

Crossrefs

Cf. A301554, A301555, A301556, A319647, A321042, A321068.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^k))^DivisorSigma[n, k], {k, 1, n}], {x, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Oct 27 2018 *)
  • PARI
    {a(n) = polcoeff(prod(k=1, n, ((1+x^k+x*O(x^n))/(1-x^k+x*O(x^n)))^sigma(k, n)), n)}

A321260 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^(sigma_n(k)-k^n).

Original entry on oeis.org

1, 0, 1, 1, 18, 2, 861, 132, 106024, 40910, 72980055, 6838271, 228282942581, 27620223647, 2050169324675668, 352809815149813, 87174966874755673105, 6798293425492905407, 18318448554980083512011863, 1187839217207171380193247, 11258918803635775614062752424535
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 01 2018

Keywords

Crossrefs

Cf. A283333, A318783, A318784, A319647, A321258, A321261.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 - x^k)^(DivisorSigma[n, k] - k^n), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[Exp[Sum[DivisorSigma[n + 1, k] x^(2 k)/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 20}]

Formula

a(n) = [x^n] exp(Sum_{k>=1} sigma_(n+1)(k)*x^(2*k)/(k*(1 - x^k))).

A321584 Number of connected (0,1)-matrices with n ones and no zero rows or columns.

Original entry on oeis.org

1, 1, 2, 6, 27, 159, 1154, 9968, 99861, 1138234, 14544650, 205927012, 3199714508, 54131864317, 990455375968, 19488387266842, 410328328297512, 9205128127109576, 219191041679766542, 5521387415218119528, 146689867860276432637, 4099255234885039058842, 120199458455807733040338
Offset: 0

Views

Author

Gus Wiseman, Nov 13 2018

Keywords

Comments

A matrix is connected if the positions in each row (or each column) of the nonzero entries form a connected hypergraph.

Examples

			The a(4) = 27 matrices:
  [1111]
.
  [111][111][111][11][110][110][101][101][100][011][011][010][001]
  [100][010][001][11][101][011][110][011][111][110][101][111][111]
.
  [11][11][11][11][10][10][10][10][01][01][01][01]
  [10][10][01][01][11][11][10][01][11][11][10][01]
  [10][01][10][01][10][01][11][11][10][01][11][11]
.
  [1]
  [1]
  [1]
  [1]

Links

Crossrefs

Cf. A007716, A007718, A049311, A056156, A101370, A104602, A120733, A283877, A319557, A319647, A319616-A319629, A321585.

Programs

  • Mathematica
    csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],Length[csm[Map[Last,GatherBy[#,First],{2}]]]==1]&]],{n,6}] (* Mathematica 7.0+ *)
  • PARI
    NonZeroCols(M)={my(C=Vec(M)); Mat(vector(#C, n, sum(k=1, n, (-1)^(n-k)*binomial(n,k)*C[k])))}
ConnectedMats(M)={my([m,n]=matsize(M), R=matrix(m,n)); for(m=1, m, for(n=1, n, R[m,n] = M[m,n] - sum(i=1, m-1, sum(j=1, n-1, binomial(m-1,i-1)*binomial(n,j)*R[i,j]*M[m-i,n-j])))); R}
seq(n)={my(M=matrix(n,n,i,j,sum(k=1, n, binomial(i*j,k)*x^k, O(x*x^n) ))); Vec(1 + vecsum(vecsum(Vec( ConnectedMats( NonZeroCols( NonZeroCols(M)~)) ))))} \\ Andrew Howroyd, Jan 17 2024

Extensions

a(7) onwards from Andrew Howroyd, Jan 17 2024

A321190 a(n) = [x^n] 1/(1 - Sum_{k>=1} k^n*x^k/(1 - x^k)).

Original entry on oeis.org

1, 1, 6, 47, 778, 25476, 1752936, 242632397, 70015221566, 41446777283255, 49999934258165654, 125272856707074638221, 641938223803783115191706, 6731818441446626626586172740, 146378489075644780343627471981694, 6505906463580477520696075719916583118
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 29 2018

Keywords

Crossrefs

Cf. A023887, A129921, A180305, A319647, A320649, A321042.

Programs

  • Maple
    seq(coeff(series((1-add(k^n*x^k/(1-x^k),k=1..n))^(-1),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 29 2018
  • Mathematica
    Table[SeriesCoefficient[1/(1 - Sum[k^n x^k/(1 - x^k), {k, 1, n}]), {x, 0, n}], {n, 0, 15}]
Table[SeriesCoefficient[1/(1 - Sum[DivisorSigma[n, k] x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 15}]
Table[SeriesCoefficient[1/(1 - Sum[Sum[j^n x^(i j), {j, 1, n}], {i, 1, n}]), {x, 0, n}], {n, 0, 15}]

Formula

a(n) = [x^n] 1/(1 - Sum_{k>=1} sigma_n(k)*x^k).
a(n) = [x^n] 1/(1 - Sum_{i>=1, j>=1} j^n*x^(i*j)).
a(n) = [x^n] 1/(1 + x * (d/dx) log(Product_{k>=1} (1 - x^k)^(k^(n-1)))).

A321264 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^J_n(k), where J_() is the Jordan function.

Original entry on oeis.org

1, 1, 4, 34, 456, 12388, 677244, 69513187, 13727785600, 5551190294478, 4378921597198116, 6705804947252051188, 21038823519531799964724, 131183284379709847290156854, 1603688086811508900855649976528, 40293997364837932973226463649637881, 2031337795407293560044987268598542021504
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 01 2018

Keywords

Links

Crossrefs

Cf. A059379, A059380, A061255, A301875, A319647, A321265.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 - x^k)^Sum[d^n MoebiusMu[k/d], {d, Divisors[k]}], {k, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Exp[Sum[Sum[Sum[d j^n MoebiusMu[d/j], {j, Divisors[d]}], {d, Divisors[k]}] x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 16}]

Formula

a(n) = [x^n] exp(Sum_{k>=1} ( Sum_{d|k} Sum_{j|d} d*j^n*mu(d/j) ) * x^k/k).
Showing 1-10 of 11 results. Next

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