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 1-4 of 4 results.

A357782 a(n) = Sum_{k=0..floor(n/3)} 2^k * Stirling2(n,3*k).

Original entry on oeis.org

1, 0, 0, 2, 12, 50, 184, 686, 2996, 16642, 110328, 784190, 5645876, 40685762, 296458344, 2226254766, 17564381332, 147289101090, 1312394060536, 12305546886398, 119906479624084, 1202273551045474, 12341175064817576, 129582557972751918, 1394497073432776756
Offset: 0

Views

Author

Seiichi Manyama, Oct 13 2022

Keywords

Links

Crossrefs

Cf. A357783, A357784.
Cf. A143815, A357831.

Programs

  • PARI
    a(n) = sum(k=0, n\3, 2^k*stirling(n, 3*k, 2));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N\3, 2^k*(exp(x)-1)^(3*k)/(3*k)!)))
  • PARI
    Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = my(v=2^(1/3), w=(-1+sqrt(3)*I)/2); round(Bell_poly(n, v)+Bell_poly(n, v*w)+Bell_poly(n, v*w^2))/3;

Formula

Let A(0)=1, B(0)=0 and C(0)=0. Let B(n+1) = Sum_{k=0..n} binomial(n,k)*A(k), C(n+1) = Sum_{k=0..n} binomial(n,k)*B(k) and A(n+1) = 2 * Sum_{k=0..n} binomial(n,k)*C(k). a(n) = A(n), A357783(n) = B(n) and A357784(n) = C(n).
Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + exp(w*x) + exp(w^2*x))/3 = 1 + x^3/3! + x^6/6! + ... . Then the e.g.f. for the sequence is F(2^(1/3) * (exp(x)-1)).
a(n) = ( Bell_n(2^(1/3)) + Bell_n(2^(1/3)*w) + Bell_n(2^(1/3)*w^2) )/3, where Bell_n(x) is n-th Bell polynomial.

A357832 a(n) = Sum_{k=0..floor((n-1)/3)} 2^k * |Stirling1(n,3*k+1)|.

Original entry on oeis.org

0, 1, 1, 2, 8, 44, 290, 2194, 18690, 177072, 1848048, 21079332, 260998584, 3487438476, 50030096844, 767092681992, 12520306878720, 216760973139072, 3967857438205320, 76575231882844056, 1553981718941428824, 33082675130470434336, 737250032464248840192
Offset: 0

Views

Author

Seiichi Manyama, Oct 14 2022

Keywords

Links

Crossrefs

Cf. A357831, A357833.

Programs

  • Mathematica
    a[n_] := With[{v = 2^(1/3), w = (-1 + Sqrt[3]*I)/2}, Round[(Pochhammer[v, n] + w^2*Pochhammer[v*w, n] + w*Pochhammer[v*w^2, n])/(3*v)]];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Oct 16 2022, after 3rd PARI code *)
  • PARI
    a(n) = sum(k=0, (n-1)\3, 2^k*abs(stirling(n, 3*k+1, 1)));
  • PARI
    my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, N\3, 2^k*(-log(1-x))^(3*k+1)/(3*k+1)!))))
  • PARI
    Pochhammer(x, n) = prod(k=0, n-1, x+k);
a(n) = my(v=2^(1/3), w=(-1+sqrt(3)*I)/2); round((Pochhammer(v, n)+w^2*Pochhammer(v*w, n)+w*Pochhammer(v*w^2, n))/(3*v));

Formula

Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w^2*exp(w*x) + w*exp(w^2*x))/3 = x + x^4/4! + x^7/7! + ... . Then the e.g.f. for the sequence is F(-2^(1/3) * log(1-x))/(2^(1/3)).
a(n) = ( (2^(1/3))_n + w^2 * (2^(1/3)*w)_n + w * (2^(1/3)*w^2)_n )/(3*2^(1/3)), where (x)_n is the Pochhammer symbol.

A357833 a(n) = Sum_{k=0..floor((n-2)/3)} 2^k * |Stirling1(n,3*k+2)|.

Original entry on oeis.org

0, 0, 1, 3, 11, 52, 304, 2114, 16992, 154626, 1568706, 17535108, 213965520, 2828584824, 40259041188, 613656673476, 9971942784132, 172071391424832, 3141974627361216, 60523400730707208, 1226519845766281008, 26084378634267048984, 580854626450078463000
Offset: 0

Views

Author

Seiichi Manyama, Oct 14 2022

Keywords

Links

Crossrefs

Cf. A357831, A357832.

Programs

  • PARI
    a(n) = sum(k=0, (n-2)\3, 2^k*abs(stirling(n, 3*k+2, 1)));
  • PARI
    my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(sum(k=0, N\3, 2^k*(-log(1-x))^(3*k+2)/(3*k+2)!))))
  • PARI
    Pochhammer(x, n) = prod(k=0, n-1, x+k);
a(n) = my(v=2^(1/3), w=(-1+sqrt(3)*I)/2); round((Pochhammer(v, n)+w*Pochhammer(v*w, n)+w^2*Pochhammer(v*w^2, n))/(3*v^2));

Formula

Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w*exp(w*x) + w^2*exp(w^2*x))/3 = x^2/2! + x^5/5! + x^8/8! + ... . Then the e.g.f. for the sequence is F(-2^(1/3) * log(1-x))/(2^(2/3)).
a(n) = ( (2^(1/3))_n + w * (2^(1/3)*w)_n + w^2 * (2^(1/3)*w^2)_n )/(3*2^(2/3)), where (x)_n is the Pochhammer symbol.

A356361 a(n) = Sum_{k=0..floor(n/3)} n^k * |Stirling1(n,3*k)|.

Original entry on oeis.org

1, 0, 0, 3, 24, 175, 1386, 12397, 125664, 1431261, 18099300, 251194911, 3788383248, 61584927495, 1072118178768, 19882255276485, 391068812992512, 8128569896422821, 177984169080865992, 4094103029211918567, 98692513234032009600, 2487731188418039207007
Offset: 0

Views

Author

Seiichi Manyama, Oct 16 2022

Keywords

Links

Crossrefs

Cf. A000407, A357683.
Cf. A356362, A356363.
Cf. A079978, A357831.

Programs

  • PARI
    a(n) = sum(k=0, n\3, n^k*abs(stirling(n, 3*k, 1)));
  • PARI
    a(n) = n!*polcoef(sum(k=0, n\3, n^k*(-log(1-x+x*O(x^n)))^(3*k)/(3*k)!), n);
  • PARI
    Pochhammer(x, n) = prod(k=0, n-1, x+k);
a(n) = my(v=n^(1/3), w=(-1+sqrt(3)*I)/2); round(Pochhammer(v, n)+Pochhammer(v*w, n)+Pochhammer(v*w^2, n))/3;

Formula

Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + exp(w*x) + exp(w^2*x))/3 = 1 + x^3/3! + x^6/6! + ... . a(n) = n! * [x^n] F(-n^(1/3) * log(1-x)).
a(n) = ( (n^(1/3))_n + (n^(1/3)*w)_n + (n^(1/3)*w^2)_n )/3, where (x)_n is the Pochhammer symbol.
Showing 1-4 of 4 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.