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 1-7 of 7 results.

A319647 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^sigma_n(k).

Original entry on oeis.org

1, 1, 6, 38, 526, 13074, 702813, 70939556, 13879861574, 5583837482767, 4393101918607162, 6717450870069292051, 21057681806321501744772, 131246096280071506595491449, 1604095619160115980216291007253, 40299198842857238408636666363954678, 2031474817845087309816967328335309651478
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 26 2018

Keywords

Links

Crossrefs

Cf. A006171, A061256, A275585, A288391, A301542, A301543, A301544, A301545, A301546, A301547, A321042.

Programs

  • Maple
    with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
      sigma[k](d), d=divisors(j))*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 26 2018
  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 - x^k)^DivisorSigma[n, k], {k, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Product[Product[1/(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^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 16}]
  • PARI
    {a(n) = polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^n))^sigma(k, n)), n)} \\ Seiichi Manyama, Oct 27 2018

Formula

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

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)}

A321068 a(n) = [x^n] Product_{k>=1} ((1 - x^k)/(1 + x^k))^sigma_n(k).

Original entry on oeis.org

1, -2, -8, -22, 294, 24982, 1372372, 10145326, -38651841784, -21995644478504, -5088041946350856, 29713279339187796814, 155715351422115081062330, 370606511915720675179342334, -12360092915168107023209454901320
Offset: 0

Views

Author

Seiichi Manyama, Oct 26 2018

Keywords

Links

Crossrefs

Cf. A320908, A320971, A320972, A321042, A321057.

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)}

A321261 a(n) = [x^n] Product_{k>=1} (1 + x^k)^(sigma_n(k)-k^n).

Original entry on oeis.org

1, 0, 1, 1, 17, 2, 859, 131, 105508, 40907, 72916903, 6834168, 228239366293, 27616985835, 2050004858009336, 352807044193881, 87173272463714343166, 6798224808203572198, 18318379579349549499397403, 1187836799227050499295342, 11258903016282277676462826232428
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 01 2018

Keywords

Crossrefs

Cf. A281268, A318844, A319107, A321042, A321258, A321260.

Programs

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

Formula

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

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)))).

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)))).

A321265 a(n) = [x^n] Product_{k>=1} (1 + x^k)^J_n(k), where J_() is the Jordan function.

Original entry on oeis.org

1, 1, 3, 33, 425, 12083, 665707, 68834806, 13654633905, 5535319947544, 4371956013518511, 6700051541666225780, 21029477920140943174285, 131152064162504305814647983, 1603485136950993248524876767297, 40291404321882574322412345562762188, 2031269423141309839019651314585293713041
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 01 2018

Keywords

Links

Crossrefs

Cf. A059379, A059380, A299069, A301876, A321042, A321264.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(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[(-1)^(k/d + 1) 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} (-1)^(k/d+1)*d*j^n*mu(d/j) ) * x^k/k).
Showing 1-7 of 7 results.

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