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.

A357720 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cos( sqrt(k) * log(1+x) ).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, -1, 0, 1, 0, -2, 3, 0, 1, 0, -3, 6, -10, 0, 1, 0, -4, 9, -18, 40, 0, 1, 0, -5, 12, -24, 60, -190, 0, 1, 0, -6, 15, -28, 60, -216, 1050, 0, 1, 0, -7, 18, -30, 40, -84, 756, -6620, 0, 1, 0, -8, 21, -30, 0, 200, -756, -1620, 46800, 0, 1, 0, -9, 24, -28, -60, 630, -3360, 13104, -14256, -365300, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2022

Keywords

Examples

			Square array begins:
  1,   1,   1,   1,   1,   1, ...
  0,   0,   0,   0,   0,   0, ...
  0,  -1,  -2,  -3,  -4,  -5, ...
  0,   3,   6,   9,  12,  15, ...
  0, -10, -18, -24, -28, -30, ...
  0,  40,  60,  60,  40,   0, ...

Links

Crossrefs

Columns k=0-4 give: A000007, (-1)^n * A003703, A357693, A357718, A357719.
Main diagonal gives A357721.
Cf. A357712, A357728.

Programs

  • PARI
    T(n, k) = sum(j=0, n\2, (-k)^j*stirling(n, 2*j, 1));
  • PARI
    T(n, k) = (-1)^n*round((prod(j=0, n-1, sqrt(k)*I+j)+prod(j=0, n-1, -sqrt(k)*I+j)))/2;

Formula

T(n,k) = Sum_{j=0..floor(n/2)} (-k)^j * Stirling1(n,2*j).
T(n,k) = (-1)^n * ( (sqrt(k) * i)_n + (-sqrt(k) * i)_n )/2, where (x)_n is the Pochhammer symbol and i is the imaginary unit.
T(0,k) = 1, T(1,k) = 0; T(n,k) = -(2*n-3) * T(n-1,k) - (n^2-4*n+4+k) * T(n-2,k).

A357703 Expansion of e.g.f. cosh( sqrt(3) * log(1-x) ).

Original entry on oeis.org

1, 0, 3, 9, 42, 240, 1614, 12474, 108900, 1059696, 11371932, 133410420, 1698541416, 23324023008, 343606235544, 5405580540360, 90445832210448, 1603781918563968, 30042007763367600, 592788643008571152, 12289695299276133024, 267079782474700715520
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2022

Keywords

Links

Crossrefs

Column k=3 of A357712.
Cf. A357615.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cosh(sqrt(3)*log(1-x)))))
  • PARI
    a(n) = sum(k=0, n\2, 3^k*abs(stirling(n, 2*k, 1)));
  • PARI
    a(n) = round((prod(k=0, n-1, sqrt(3)+k)+prod(k=0, n-1, -sqrt(3)+k)))/2;
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=2, n, v[i+1]=(2*i-3)*v[i]-(i^2-4*i+1)*v[i-1]); v;

Formula

a(n) = Sum_{k=0..floor(n/2)} 3^k * |Stirling1(n,2*k)|.
a(n) = ( (sqrt(3))_n + (-sqrt(3))_n )/2, where (x)_n is the Pochhammer symbol.
a(0) = 1, a(1) = 0; a(n) = (2*n-3) * a(n-1) - (n^2-4*n+1) * a(n-2).

A357711 Expansion of e.g.f. cosh( 2 * log(1-x) ).

Original entry on oeis.org

1, 0, 4, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800, 3113510400, 43589145600, 653837184000, 10461394944000, 177843714048000, 3201186852864000, 60822550204416000, 1216451004088320000, 25545471085854720000, 562000363888803840000
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2022

Keywords

Links

Crossrefs

Column k=4 of A357712.
Cf. A065143.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(cosh(2*log(1-x))))
  • PARI
    a(n) = sum(k=0, n\2, 4^k*abs(stirling(n, 2*k, 1)));
  • PARI
    a(n) = (prod(k=0, n-1, 2+k)+prod(k=0, n-1, -2+k))/2;
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=2, n, v[i+1]=(2*i-3)*v[i]-(i^2-4*i)*v[i-1]); v;

Formula

a(n) = Sum_{k=0..floor(n/2)} 4^k * |Stirling1(n,2*k)|.
a(n) = ( (2)_n + (-2)_n )/2, where (x)_n is the Pochhammer symbol.
a(n) = (n+1)!/2 for n > 2.
a(0) = 1, a(1) = 0; a(n) = (2*n-3) * a(n-1) - (n^2-4*n) * a(n-2).
Showing 1-3 of 3 results.

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