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.

A357728 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) * (exp(x) - 1) ).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, -1, 0, 1, 0, -2, -3, 0, 1, 0, -3, -6, -6, 0, 1, 0, -4, -9, -10, -5, 0, 1, 0, -5, -12, -12, 10, 33, 0, 1, 0, -6, -15, -12, 45, 190, 266, 0, 1, 0, -7, -18, -10, 100, 465, 1106, 1309, 0, 1, 0, -8, -21, -6, 175, 852, 2394, 4438, 4905, 0, 1, 0, -9, -24, 0, 270, 1345, 4004, 7827, 9978, 11516, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 11 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, -6, -10, -12, -12, -10, ...
  0, -5,  10,  45, 100, 175, ...

Links

Crossrefs

Columns k=0-4 give: A000007, A121867, A357725, A357726, A357727.
Main diagonal gives A357729.
Cf. A357681, A357720.

Programs

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

Formula

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

A357719 Expansion of e.g.f. cos( 2 * log(1+x) ).

Original entry on oeis.org

1, 0, -4, 12, -28, 40, 200, -3360, 35680, -357120, 3644800, -38896000, 437756800, -5206406400, 65372153600, -864339840000, 11991424640000, -173800340480000, 2617640829440000, -40693929269760000, 647089190924800000, -10383194262604800000
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2022

Keywords

Links

Crossrefs

Column k=4 of A357720.
Cf. A357711, A357727.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(cos(2*log(1+x))))
  • PARI
    a(n) = sum(k=0, n\2, (-4)^k*stirling(n, 2*k, 1));
  • PARI
    a(n) = (-1)^n*(prod(k=0, n-1, 2*I+k)+prod(k=0, n-1, -2*I+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+8)*v[i-1]); v;

Formula

a(n) = Sum_{k=0..floor(n/2)} (-4)^k * Stirling1(n,2*k).
a(n) = (-1)^n * ( (2 * i)_n + (-2 * i)_n )/2, where (x)_n is the Pochhammer symbol and i is the imaginary unit.
a(0) = 1, a(1) = 0; a(n) = -(2*n-3) * a(n-1) - (n^2-4*n+8) * a(n-2).

A357718 Expansion of e.g.f. cos( sqrt(3) * log(1+x) ).

Original entry on oeis.org

1, 0, -3, 9, -24, 60, -84, -756, 13104, -157248, 1795248, -20900880, 254007936, -3250473408, 43922668608, -626830626240, 9437477107968, -149644407564288, 2493958878657792, -43592393744250624, 797394015216175104, -15230735270523601920
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2022

Keywords

Links

Crossrefs

Column k=3 of A357720.
Cf. A357703, A357726.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cos(sqrt(3)*log(1+x)))))
  • PARI
    a(n) = sum(k=0, n\2, (-3)^k*stirling(n, 2*k, 1));
  • PARI
    a(n) = (-1)^n*round((prod(k=0, n-1, sqrt(3)*I+k)+prod(k=0, n-1, -sqrt(3)*I+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+7)*v[i-1]); v;

Formula

a(n) = Sum_{k=0..floor(n/2)} (-3)^k * Stirling1(n,2*k).
a(n) = (-1)^n * ( (sqrt(3) * i)_n + (-sqrt(3) * i)_n )/2, where (x)_n is the Pochhammer symbol and i is the imaginary unit.
a(0) = 1, a(1) = 0; a(n) = -(2*n-3) * a(n-1) - (n^2-4*n+7) * a(n-2).

A357721 a(n) = Sum_{k=0..floor(n/2)} (-n)^k * Stirling1(n,2*k).

Original entry on oeis.org

1, 0, -2, 9, -28, 0, 1200, -16464, 167904, -1393200, 7429240, 43124400, -2404571904, 55590286752, -1027511503200, 16489054310400, -222885864448000, 1994839594780032, 14489184835474272, -1470395490046560000, 54581408106475622400, -1608207353670788640000
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2022

Keywords

Links

Crossrefs

Main diagonal of A357720.
Cf. A357683, A357729.

Programs

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

Formula

a(n) = n! * [x^n] cos( sqrt(n) * log(1+x) ).
a(n) = (-1)^n * ( (sqrt(n) * i)_n + (-sqrt(n) * i)_n )/2, where (x)_n is the Pochhammer symbol and i is the imaginary unit.

A357693 Expansion of e.g.f. cos( sqrt(2) * log(1+x) ).

Original entry on oeis.org

1, 0, -2, 6, -18, 60, -216, 756, -1620, -14256, 349272, -5452920, 78885576, -1143659088, 17074183104, -265437239760, 4316991698448, -73572489226368, 1314108286270560, -24584195654596512, 481215937895868384, -9843358555320333120, 210128893733994567552
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2022

Keywords

Links

Crossrefs

Column k=2 of A357720.
Cf. A003703, A357718, A357719.
Cf. A357725.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Cos[Sqrt[2]Log[1+x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 04 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cos(sqrt(2)*log(1+x)))))
  • PARI
    a(n) = sum(k=0, n\2, (-2)^k*stirling(n, 2*k, 1));
  • PARI
    a(n) = (-1)^n*round((prod(k=0, n-1, sqrt(2)*I+k)+prod(k=0, n-1, -sqrt(2)*I+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+6)*v[i-1]); v;

Formula

a(n) = Sum_{k=0..floor(n/2)} (-2)^k * Stirling1(n,2*k).
a(n) = (-1)^n * ( (sqrt(2) * i)_n + (-sqrt(2) * i)_n )/2, where (x)_n is the Pochhammer symbol and i is the imaginary unit.
a(0) = 1, a(1) = 0; a(n) = -(2*n-3) * a(n-1) - (n^2-4*n+6) * a(n-2).
Showing 1-5 of 5 results.

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