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 1-2 of 2 results.

A138461 Inverse binomial transform of A000957.

Original entry on oeis.org

0, 1, -2, 4, -6, 11, -14, 29, -26, 85, -12, 320, 312, 1639, 3190, 10484, 25822, 75005, 200488, 564662, 1555804, 4363139, 12184456, 34267931, 96435100, 272390561, 770734846, 2186278294, 6213111234
Offset: 0

Views

Author

N. J. A. Sloane, May 08 2008

Keywords

Links

Crossrefs

Cf. A000957, A138415.

Programs

  • Mathematica
    Table[Sum[(-1)^(n-k) * Binomial[n, k]*(2^k * (2*k-1)!! * Hypergeometric2F1Regularized[2, 2*k+1, k+2, -1] - 3*(-1)^k/2^(k+1)), {k, 1, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 30 2017 *)

Formula

From Vaclav Kotesovec, Oct 30 2017: (Start)
Recurrence: 2*n*a(n) = -(n+6)*a(n-1) + (13*n - 33)*a(n-2) + 3*(7*n - 18)*a(n-3) + 9*(n-3)*a(n-4).
a(n) ~ 3^(n - 1/2) / (8 * sqrt(Pi) * n^(3/2)). (End)

A373746 Triangle read by rows: the almost-Riordan array ( 1/(1-x) | 2/((1-x)*(1+x+sqrt(5*x^2-6*x+1))), (1-3*x-sqrt(5*x^2-6*x+1))/(2x) ).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 5, 1, 1, 10, 20, 8, 1, 1, 31, 78, 45, 11, 1, 1, 110, 310, 224, 79, 14, 1, 1, 421, 1264, 1061, 475, 122, 17, 1, 1, 1686, 5274, 4922, 2608, 858, 174, 20, 1, 1, 6961, 22430, 22648, 13604, 5356, 1400, 235, 23, 1, 1, 29392, 96899, 103978, 68816, 31072, 9791, 2128, 305, 26, 1
Offset: 0

Views

Author

Stefano Spezia, Jun 16 2024

Keywords

Comments

In He and Słowik, there is a typing error since T(5,1) is equal to 31 and not to 421.

Examples

			The triangle begins as:
  1;
  1,   1;
  1,   2,   1;
  1,   4,   5,   1;
  1,  10,  20,   8,  1;
  1,  31,  78,  45, 11,  1;
  1, 110, 310, 224, 79, 14, 1;
  ...

Links

Crossrefs

Cf. A000012 (k=0 and n=k), A016789, A138415 (k=1).

Programs

  • Mathematica
    T[n_, 0]:=1; T[n_, k_]:=SeriesCoefficient[2/((1-x)(1+x+Sqrt[5x^2-6x+1]))((1-3x-Sqrt[5x^2-6x+1])/(2x))^(k-1), {x, 0, n-1}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Formula

T(n,n-1) = A016789(n-2).
Showing 1-2 of 2 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.