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 1-3 of 3 results.

A347915 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^exp(x).

Original entry on oeis.org

1, 1, 4, 24, 150, 1235, 11725, 126987, 1512084, 20313897, 296921623, 4700713787, 80221988726, 1468879687145, 28661345212981, 594457831566757, 13027193829914920, 301079987772726257, 7318797530268562203, 186496088631167771143, 4971371842655844396298, 138384071439982000722737
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2021

Keywords

Crossrefs

Cf. A000593, A088311, A346547, A347916, A354507.
Cf. A354503, A354504.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[(1 + x^k)^Exp[x], {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 17 2022 *)
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1+x^k)^exp(x))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(exp(x)*sum(k=1, N, sigma(k>>valuation(k, 2))*x^k/k))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(exp(x)*sum(k=1, N, x^k/(k*(1-x^(2*k)))))))
  • PARI
    a354507(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d)/(k*(n-k)!));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354507(j)*binomial(i-1, j-1)*v[i-j+1])); v; \\ Seiichi Manyama, Aug 16 2022

Formula

E.g.f.: exp( exp(x) * Sum_{k>=1} A000593(k)*x^k/k ).
E.g.f.: exp( exp(x) * Sum_{k>=1} x^k/(k*(1 - x^(2*k))) ).
a(0) = 1; a(n) = Sum_{k=1..n} A354507(k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Aug 16 2022

A354506 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) )/(k * (n-k)!).

Original entry on oeis.org

1, 2, 7, 14, 63, 284, 2385, 3940, 87717, 940126, 12743267, 30055618, 562302323, 9005878920, 423435780989, 2080544097000, 24457758561001, 444510436079706, 17533073308723423, 46973556239255702, 7501223613055891783, 178483805340054632084, 4396051786608296882889, -31788150263554644516724
Offset: 1

Views

Author

Seiichi Manyama, Aug 15 2022

Keywords

Crossrefs

Cf. A354507, A354508.
Cf. A048272, A356389.

Programs

  • PARI
    a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1))/(k*(n-k)!));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=1, N, log(1+x^k)/k)))

Formula

a(n) = n! * Sum_{k=1..n} A048272(k)/(k * (n-k)!).
E.g.f.: exp(x) * Sum_{k>0} log(1 + x^k)/k.

A354508 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * d^2 )/(k * (n-k)!).

Original entry on oeis.org

1, 5, 32, 168, 1189, 8785, 77384, 646296, 7306737, 79997893, 1005481784, 12518370128, 184109233125, 2671256865121, 47934480000112, 754158322407248, 13813898274148737, 262680987222463269, 5518034466415262320, 107988236156057411096, 2605128008760639636677
Offset: 1

Views

Author

Seiichi Manyama, Aug 15 2022

Keywords

Crossrefs

Cf. A354506, A354507.
Cf. A078306, A356391.

Programs

  • PARI
    a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^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

a(n) = n! * Sum_{k=1..n} A078306(k)/(k * (n-k)!).
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).
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.