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 1-5 of 5 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).

A357725 Expansion of e.g.f. cos( sqrt(2) * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, -2, -6, -10, 10, 190, 1106, 4438, 9978, -35250, -666622, -5657370, -35308182, -155215970, -128513870, 7051468022, 105057922906, 1042016038254, 8053738122466, 44608555196294, 48639210067658, -3200193654245442, -60669816166988654, -769281697485061994
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2022

Keywords

Links

Crossrefs

Column k=2 of A357728.
Cf. A121867, A264036, A357736.

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/2)} (-2)^k * Stirling2(n,2*k).
a(n) = 1; a(n) = -2 * Sum_{k=0..n-1} binomial(n-1, k) * A357736(k).
a(n) = ( Bell_n(sqrt(2) * i) + Bell_n(-sqrt(2) * i) )/2, where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.

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.

A357727 Expansion of e.g.f. cos( 2 * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, -4, -12, -12, 100, 852, 4004, 9940, -36828, -726316, -6174300, -35968812, -109708508, 702818004, 16677814436, 188794428628, 1542659688996, 8359981681364, -3068614764636, -868989327994668, -15076627082974940, -179727483880747308
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2022

Keywords

Links

Crossrefs

Column k=4 of A357728.
Cf. A065143, A357719, A357738.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(cos(2*(exp(x)-1))))
  • PARI
    a(n) = sum(k=0, n\2, (-4)^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, 2*I)+Bell_poly(n, -2*I)))/2;

Formula

a(n) = Sum_{k=0..floor(n/2)} (-4)^k * Stirling2(n,2*k).
a(n) = 1; a(n) = -4 * Sum_{k=0..n-1} binomial(n-1, k) * A357738(k).
a(n) = ( Bell_n(2 * i) + Bell_n(-2 * i) )/2, where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.

A357729 a(n) = Sum_{k=0..floor(n/2)} (-n)^k * Stirling2(n,2*k).

Original entry on oeis.org

1, 0, -2, -9, -12, 175, 1938, 9506, -24248, -1065663, -12021610, -56195425, 677072220, 19979234080, 251733387514, 1135594212255, -29317384858352, -901607623649489, -13233854770928514, -68574233644270566, 2258648937829442660, 81748108921355457777
Offset: 0

Views

Author

Seiichi Manyama, Oct 11 2022

Keywords

Links

Crossrefs

Main diagonal of A357728.
Cf. A357682, A357721.

Programs

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

Formula

a(n) = n! * [x^n] cos( sqrt(n) * (exp(x) - 1) ).
a(n) = ( Bell_n(sqrt(n) * i) + Bell_n(-sqrt(n) * i) )/2, where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.
Showing 1-5 of 5 results.

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