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.

A356297 a(n) = n! * Sum_{k=1..n} sigma_0(k)/k.

Original entry on oeis.org

1, 4, 16, 82, 458, 3228, 24036, 212448, 2032992, 21781440, 246853440, 3201742080, 42580650240, 621037186560, 9664270963200, 161166707251200, 2781679603046400, 52204357423411200, 1004687538456268800, 20823621371578368000, 447027656835852288000
Offset: 1

Views

Author

Seiichi Manyama, Aug 03 2022

Keywords

Crossrefs

Cf. A000005, A006218, A356010, A356298, A356323.

Programs

  • Mathematica
    Table[n! * Sum[DivisorSigma[0, k]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)
  • PARI
    a(n) = n!*sum(k=1, n, sigma(k, 0)/k);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-x^k)/k)/(1-x)))

Formula

E.g.f.: -(1/(1-x)) * Sum_{k>0} log(1 - x^k)/k.
a(n) ~ n! * (log(n)^2/2 + 2*gamma*log(n) + gamma^2 - 2*sg1), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Aug 07 2022

A356337 Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k)^k )^(1/(1-x)).

Original entry on oeis.org

1, 1, 8, 63, 644, 7610, 107994, 1713726, 30671024, 603160344, 12974475240, 301879678320, 7561610279112, 202437968475288, 5769455216675136, 174234738889383480, 5556311629901103360, 186482786151757707840, 6568881383985687359424, 242221409390815100812224
Offset: 0

Views

Author

Seiichi Manyama, Aug 04 2022

Keywords

Crossrefs

Cf. A000219, A001157, A356298, A356335, A356336.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Product[1/((1-x^k)^k)^(1/(1-x)),{k,nn}],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 06 2023 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((1/prod(k=1, N, (1-x^k)^k))^(1/(1-x))))
  • PARI
    a356298(n) = n!*sum(k=1, n, sigma(k, 2)/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356298(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A356298(k) * binomial(n-1,k-1) * a(n-k).

A356391 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d + 1) * d^2 ) /k.

Original entry on oeis.org

1, 5, 35, 206, 1654, 13524, 130668, 1262064, 15027696, 178581600, 2407111200, 33276182400, 514020643200, 8130342124800, 144621487584000, 2537556118272000, 49206063078144000, 982811803276800000, 20991083543732736000, 454612169591580672000, 10763306565511514112000
Offset: 1

Views

Author

Seiichi Manyama, Aug 05 2022

Keywords

Links

Crossrefs

Cf. A356389, A356390.
Cf. A078306, A356298, A356394.

Programs

  • Mathematica
    Table[n! * Sum[Sum[(-1)^(k/d + 1)*d^2, {d, Divisors[k]}]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)
  • PARI
    a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^2)/k);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, (-x)^k/(k*(1-x^k)^2))/(1-x)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k*log(1+x^k))/(1-x)))

Formula

a(n) = n! * Sum_{k=1..n} A078306(k)/k.
E.g.f.: -(1/(1-x)) * Sum_{k>0} (-x)^k/(k * (1 - x^k)^2).
E.g.f.: (1/(1-x)) * Sum_{k>0} k * log(1 + x^k).
a(n) ~ n! * n^2 * 3 * zeta(3) / 8. - Vaclav Kotesovec, Aug 07 2022

A353992 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} d^(k/d + 1) )/k.

Original entry on oeis.org

1, 7, 41, 314, 2194, 22764, 195348, 2374224, 27940176, 384636960, 4673720160, 95522440320, 1323221996160, 23481816503040, 489968947641600, 10853692580505600, 190580382936115200, 5408424680491929600, 105077728210820198400, 3399507785578641408000
Offset: 1

Views

Author

Seiichi Manyama, Aug 06 2022

Keywords

Crossrefs

Cf. A078308, A353993, A356010, A356298, A356406.

Programs

  • Mathematica
    a[n_] := n! * Sum[DivisorSum[k, #^(k/# + 1) &]/k, {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Aug 06 2022 *)
  • PARI
    a(n) = n!*sum(k=1, n, sumdiv(k, d, d^(k/d+1))/k);
  • PARI
    a(n) = n!*sum(k=1, n, sumdiv(k, d, (k/d)^d/d));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-k*x^k))/(1-x)))

Formula

a(n) = n! * Sum_{k=1..n} A078308(k)/k.
a(n) = n! * Sum_{k=1..n} Sum_{d|k} (k/d)^d / d.
E.g.f.: -(1/(1-x)) * Sum_{k>0} log(1 - k * x^k).

A356323 a(n) = n! * Sum_{k=1..n} sigma_3(k)/k.

Original entry on oeis.org

1, 11, 89, 794, 6994, 72204, 753108, 8973264, 111281616, 1524322080, 21601104480, 340803192960, 5483287025280, 96044874750720, 1748238132614400, 34093033838438400, 682396164763084800, 14706429413353574400, 323342442475011993600, 7585740483060676608000
Offset: 1

Views

Author

Seiichi Manyama, Aug 03 2022

Keywords

Crossrefs

Cf. A001158, A064603, A356010, A356297, A356298.

Programs

  • Mathematica
    Table[n! * Sum[DivisorSigma[3, k]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)
  • PARI
    a(n) = n!*sum(k=1, n, sigma(k, 3)/k);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k*(1+x^k)/(k*(1-x^k)^3))/(1-x)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, k^2*log(1-x^k))/(1-x)))

Formula

E.g.f.: (1/(1-x)) * Sum_{k>0} x^k * (1 + x^k)/(k * (1 - x^k)^3).
E.g.f.: -(1/(1-x)) * Sum_{k>0} k^2 * log(1 - x^k).
a(n) ~ n! * Pi^4 * n^3 / 270. - Vaclav Kotesovec, Aug 07 2022

A356600 a(n) = n! * Sum_{k=1..n} sigma_2(k)/(k * (n-k)!).

Original entry on oeis.org

1, 7, 38, 240, 1509, 12115, 96326, 929432, 9421089, 108909943, 1249105054, 17862483320, 241674418101, 3676733397363, 59149265744302, 1058605924855568, 18041587282787489, 363409114370324295, 6970858463185187062, 153017341796727034336, 3360005220780469981157
Offset: 1

Views

Author

Seiichi Manyama, Aug 15 2022

Keywords

Comments

The average value of a(n) is zeta(3) * exp(1) * n * n!. - Vaclav Kotesovec, Aug 17 2022

Crossrefs

Cf. A002745, A002746, A356589.
Cf. A356298.

Programs

  • Mathematica
    Table[n! * Sum[DivisorSigma[2, k]/(k * (n-k)!), {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 17 2022 *)
  • PARI
    a(n) = n!*sum(k=1, n, sigma(k, 2)/(k*(n-k)!));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=1, N, x^k/(k*(1-x^k)^2))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x)*sum(k=1, N, k*log(1-x^k))))

Formula

E.g.f.: exp(x) * Sum_{k>0} x^k/(k * (1 - x^k)^2).
E.g.f.: -exp(x) * Sum_{k>0} k * log(1 - x^k).

A356485 a(n) = n! * Sum_{k=1..n} A000010(k)/k.

Original entry on oeis.org

1, 3, 13, 64, 416, 2736, 23472, 207936, 2113344, 22584960, 284722560, 3576337920, 52240412160, 768727895040, 12228344755200, 206114911027200, 3838718125670400, 71231050830643200, 1468632692485324800, 30345814652977152000, 666456931810639872000, 15172961921551171584000
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 09 2022

Keywords

Links

Crossrefs

Cf. A000010, A002088, A011755, A356297, A356010, A356298, A356323.

Programs

  • Mathematica
    Table[n! * Sum[EulerPhi[k]/k, {k, 1, n}], {n, 1, 25}]
  • PARI
    a(n) = n!*sum(k=1, n, eulerphi(k)/k); \\ Michel Marcus, Aug 09 2022

Formula

a(n) ~ n! * 6*n/Pi^2.
Showing 1-7 of 7 results.

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

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