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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A384216 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = n! * [x^n] (1 + k*x)^(n/k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 0, 6, 1, 1, -2, -3, 24, 1, 1, -4, 0, 0, 120, 1, 1, -6, 15, 40, 45, 720, 1, 1, -8, 42, 0, -280, 0, 5040, 1, 1, -10, 81, -264, -1155, 0, -1575, 40320, 1, 1, -12, 132, -896, 0, 20160, 24640, 0, 362880, 1, 1, -14, 195, -2040, 8645, 57456, -208845, -291200, 99225, 3628800
Offset: 0

Views

Author

Seiichi Manyama, May 22 2025

Keywords

Examples

			Square array begins:
    1,  1,    1,     1,     1,    1,       1, ...
    1,  1,    1,     1,     1,    1,       1, ...
    2,  0,   -2,    -4,    -6,   -8,     -10, ...
    6, -3,    0,    15,    42,   81,     132, ...
   24,  0,   40,     0,  -264, -896,   -2040, ...
  120, 45, -280, -1155,     0, 8645,   33120, ...
  720,  0,    0, 20160, 57456,    0, -459360, ...

Links

Crossrefs

Columns k=3..5 give A282627(n+1)/2, A383996, A383997.
Cf. A303489.

Programs

  • PARI
    a(n, k) = prod(j=0, n-1, n-k*j);

Formula

A(n,k) = Product_{j=0..n-1} (n-k*j).
A(n,k) = k^n * FallingFactorial(n/k,n).
A(k*n,k) = 0 for n > 0 and k > 1.

A282626 Exponential expansion of the real root y = y(x) of y^3 - 3*x*y - 1.

Original entry on oeis.org

1, 1, 0, -2, 8, 0, -320, 2800, 0, -344960, 4659200, 0, -1172864000, 21423001600, 0, -9117844736000, 209945415680000, 0, -135381758640128000, 3761801958154240000, 0, -3421097040836034560000, 111349337961365504000000, 0, -135776499356700539617280000
Offset: 0

Views

Author

Wolfdieter Lang, Mar 04 2017

Keywords

Comments

This is an example of an application of Ramanujan's Master theorem for definite integrals; see eq. (B) on p. 186 of the Hardy reference. This application is given under (ii) on pp. 194-195; here with r = 1, p = 1, q = 2, and x and a there are y and x here, respectively.
The general formula for the exponential expansion of the r-th power of the solution y=y(x) of y^q - q*x*y - 1 = 0 which starts with y(0) = 1 is y(x)^r = Sum_{n>=0} lambda(n;r,q,p)*x^n/n! with lambda(0;r,q,p) = 1, lambda(1;r,q,p) = r and lambda(n;r,q,p) = r*Product_{j=1..n-1} (r + n*p - q*j) for n >= 2. Hardy gives a convergence condition for theorem (B) on p. 189: the class K(A,P,delta) for phi(u) = lambda(u) / Gamma(1+u), u complex, here for lambda(u) = lambda(u;r,q,p).

References

  • G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, ch, XI, pp. 186-211.

Links

Crossrefs

Cf. A282627.

Programs

  • Magma
    [1,1] cat [(&*[n+1-3*j: j in [1..(n-1)]]): n in [2..25]]; // G. C. Greubel, Mar 29 2019
  • Mathematica
    Table[Product[n+1-3*j, {j,1,n-1}], {n,0,25}] (* G. C. Greubel, Mar 29 2019 *)
  • PARI
    vector(25, n, n--; prod(j=1,n-1, (n+1-3*j))) \\ G. C. Greubel, Mar 29 2019
  • Sage
    [1] + [product(n+1-3*j for j in (1..(n-1))) for n in (1..25)] # G. C. Greubel, Mar 29 2019

Formula

a(n) = Product_{j=1..n-1} (n + 1 - 3*j), n >=0 (empty product = 1).
E.g.f.: ((1 + sqrt(1-4*x^3))/2)^(1/3) + x/((1 + sqrt(1-4*x^3))/2)^(1/3).
E.g.f.: ((1 + sqrt(1-4*x^3))/2)^(1/3) + ((1 - sqrt(1-4*x^3))/2)^(1/3).

Extensions

More terms from G. C. Greubel, Mar 29 2019
Showing 1-2 of 2 results.

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