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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A066045 Triangle T(n,k) defined by Sum_{1<=k<=n} T(n,k)*u^k*t^n/n! = exp(((1-t)*(1-t^2)*(1-t^3)...)^(-u)-1).

Original entry on oeis.org

1, 3, 2, 8, 18, 5, 42, 118, 90, 15, 144, 900, 1075, 450, 52, 1440, 6788, 12375, 8475, 2340, 203, 5760, 61824, 141470, 140175, 63700, 12789, 877, 75600, 586584, 1700580, 2218335, 1441440, 474614, 73668, 4140, 524160, 6064416, 21677980
Offset: 1

Views

Author

Vladeta Jovovic, Dec 29 2001

Keywords

Examples

			Triangle begins:
  [1],
  [3, 2],
  [8, 18, 5],
  [42, 118, 90, 15],
  [144, 900, 1075, 450, 52],
  ...

Crossrefs

Cf. A058892 (row sums), A000110, A008298, A038048.

A318580 Expansion of e.g.f. exp(-1 + Product_{k>=1} 1/(1 - x^k)^k).

Original entry on oeis.org

1, 1, 7, 55, 601, 7561, 116191, 1999327, 39267985, 850964401, 20332107991, 527930427751, 14838001344937, 447653776595065, 14440021169407471, 495398956418435791, 18012260306904120481, 691502230924473978337, 27948692251661337581095, 1185878351946613955122711
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 29 2018

Keywords

Crossrefs

Cf. A000219, A001157, A058892, A318250.

Programs

  • Maple
    seq(n!*coeff(series(exp(-1+mul(1/(1-x^k)^k,k=1..100)),x=0,20),x,n),n=0..19); # Paolo P. Lava, Jan 09 2019
  • Mathematica
    nmax = 19; CoefficientList[Series[Exp[-1 + Product[1/(1 - x^k)^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[Exp[-1 + Exp[Sum[DivisorSigma[2, k] x^k/k, {k, 1, nmax}]]], {x, 0, nmax}], x] Range[0, nmax]!
p[n_] := p[n] = Sum[DivisorSigma[2, k] p[n - k], {k, n}]/n; p[0] = 1; a[n_] := a[n] = Sum[p[k] k! Binomial[n - 1, k - 1] a[n - k], {k, n}]; a[0] = 1; Table[a[n], {n, 0, 19}]

Formula

E.g.f.: exp(-1 + exp(Sum_{k>=1} sigma_2(k)*x^k/k)).
E.g.f.: A(x) = exp(B(x) - 1), where B(x) = o.g.f. of A000219.
a(0) = 1; a(n) = Sum_{k=1..n} A000219(k)*k!*binomial(n-1,k-1)*a(n-k).

A320899 Expansion of e.g.f. exp(1/theta_4(x) - 1), where theta_4() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 12, 104, 1120, 14592, 221824, 3835904, 74262528, 1589016320, 37181031424, 943547716608, 25791165349888, 754934109863936, 23547020011929600, 779291847538638848, 27263652732032843776, 1005002283128197349376, 38921431600215853760512, 1579513585265275661189120
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 23 2018

Keywords

Crossrefs

Cf. A015128, A058892, A293840.

Programs

  • Maple
    seq(coeff(series(factorial(n)*(exp(-1+mul((1+x^k)/(1-x^k),k=1..n))),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 23 2018
  • Mathematica
    nmax = 19; CoefficientList[Series[Exp[1/EllipticTheta[4, 0, x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Sum[PartitionsP[k - j] PartitionsQ[j], {j, 0, k}] k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 19}]

Formula

E.g.f.: exp(-1 + Product_{k>=1} (1 + x^k)/(1 - x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A015128(k)*k!*binomial(n-1,k-1)*a(n-k).
Previous Showing 11-13 of 13 results.

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