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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A357572 Expansion of e.g.f. sinh(sqrt(3) * (exp(x)-1)) / sqrt(3).

Original entry on oeis.org

0, 1, 1, 4, 19, 85, 406, 2191, 13105, 84190, 573121, 4127521, 31434184, 252388957, 2126998693, 18740283556, 172134162631, 1644920020417, 16324076578870, 167938152551491, 1787952325142341, 19667748794844550, 223217829954224029, 2610546296216999197
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2022

Keywords

Links

Crossrefs

Cf. A024429, A264037, A357598.
Cf. A027710, A357615, A357737.

Programs

  • PARI
    a(n) = sum(k=0, (n-1)\2, 3^k*stirling(n, 2*k+1, 2));
  • PARI
    Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, sqrt(3))-Bell_poly(n, -sqrt(3)))/(2*sqrt(3)));

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} 3^k * Stirling2(n,2*k+1).
a(n) = ( Bell_n(sqrt(3)) - Bell_n(-sqrt(3)) )/(2 * sqrt(3)), where Bell_n(x) is n-th Bell polynomial.
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357615(k).

A357726 Expansion of e.g.f. cos( sqrt(3) * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, -3, -9, -12, 45, 465, 2394, 7827, 639, -250410, -2588553, -17773635, -84525480, -105849399, 3569654115, 56100280308, 561682625769, 4227837863181, 20472943653306, -38990802816489, -2621206974761253, -42512769453705474, -495174030273565173
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2022

Keywords

Links

Crossrefs

Column k=3 of A357728.
Cf. A357615, A357737.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Cos[Sqrt[3](Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 20 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cos(sqrt(3)*(exp(x)-1)))))
  • PARI
    a(n) = sum(k=0, n\2, (-3)^k*stirling(n, 2*k, 2));
  • PARI
    Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, sqrt(3)*I)+Bell_poly(n, -sqrt(3)*I)))/2;

Formula

a(n) = Sum_{k=0..floor(n/2)} (-3)^k * Stirling2(n,2*k).
a(n) = 1; a(n) = -3 * Sum_{k=0..n-1} binomial(n-1, k) * A357737(k).
a(n) = ( Bell_n(sqrt(3) * i) + Bell_n(-sqrt(3) * i) )/2, where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.
Showing 1-2 of 2 results.

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