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.

A128417 Number triangle T(n,k) = 2^(n-k)*C(2*n,n-k).

Original entry on oeis.org

1, 4, 1, 24, 8, 1, 160, 60, 12, 1, 1120, 448, 112, 16, 1, 8064, 3360, 960, 180, 20, 1, 59136, 25344, 7920, 1760, 264, 24, 1, 439296, 192192, 64064, 16016, 2912, 364, 28, 1, 3294720, 1464320, 512512, 139776, 29120, 4480, 480, 32, 1
Offset: 0

Views

Author

Paul Barry, Mar 02 2007

Keywords

Comments

Inverse of A128414. Row sums are A128418. Diagonal sums are A128419.

Examples

			Triangle begins:
  1,
  4, 1,
  24, 8, 1,
  160, 60, 12, 1,
  1120, 448, 112, 16, 1,
  8064, 3360, 960, 180, 20, 1,
  59136, 25344, 7920, 1760, 264, 24, 1,
  439296, 192192, 64064, 16016, 2912, 364, 28, 1
  ...

Links

Crossrefs

Cf. A128413.

Programs

  • Mathematica
    Flatten[Table[2^(n-k) Binomial[2n,n-k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Nov 02 2011 *)

Formula

Riordan array (1/sqrt(1-8*x),(1-4*x-sqrt(1-8*x))/(8*x)).
T(n,k) = 2^(n-k)*A094527(n,k).

A128414 Riordan array ((1-2x)/(1+2x),x/(1+2x)^2).

Original entry on oeis.org

1, -4, 1, 8, -8, 1, -16, 36, -12, 1, 32, -128, 80, -16, 1, -64, 400, -400, 140, -20, 1, 128, -1152, 1680, -896, 216, -24, 1, -256, 3136, -6272, 4704, -1680, 308, -28, 1, 512, -8192, 21504, -21504, 10560, -2816
Offset: 0

Views

Author

Paul Barry, Mar 02 2007

Keywords

Comments

Inverse is A128417. Row sums are A128415. Diagonal sums are A128416.

Examples

			Triangle begins
1;
-4, 1;
8, -8, 1;
-16, 36, -12, 1;
32, -128, 80, -16, 1;
-64, 400, -400, 140, -20, 1;
128, -1152, 1680, -896, 216, -24, 1;
-256, 3136, -6272, 4704, -1680, 308, -28, 1;
512, -8192, 21504, -21504, 10560, -2816, 416, -32, 1;

Crossrefs

Cf. A128413.

Formula

T(n,k) = 2^(n-k)*A110162(n,k). - Philippe Deléham, Jan 18 2014
T(n,k) = T(n-1,k-1) - 4*T(n-1,k) - 4*T(n-2,k), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = -4, T(2,0) = 8, T(2,1) = -8, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 18 2014

A128412 A scaled version of the coefficient array for orthogonal polynomials defined by C(2n,n).

Original entry on oeis.org

1, -2, 1, 4, -8, 1, -8, 36, -12, 1, 16, -128, 80, -16, 1, -32, 400, -400, 140, -20, 1, 64, -1152, 1680, -896, 216, -24, 1, -128, 3136, -6272, 4704, -1680, 308, -28, 1, 256, -8192, 21504, -21504, 10560, -2816
Offset: 0

Views

Author

Paul Barry, Mar 02 2007

Keywords

Comments

Inverse is A128413.

Examples

			Triangle begins:
  1,
  -2, 1,
  4, -8, 1,
  -8, 36, -12, 1,
  16, -128, 80, -16, 1,
  -32, 400, -400, 140, -20, 1,
  64, -1152, 1680, -896, 216, -24, 1,
  -128, 3136, -6272, 4704, -1680, 308, -28, 1,
  256, -8192, 21504, -21504, 10560, -2816, 416, -32, 1,
  -512, 20736, -69120, 88704, -57024, 20592, -4368, 540, -36, 1,
  1024, -51200, 211200, -337920, 274560, -128128, 36400, -6400, 680, -40, 1

Formula

Number triangle T(n,k)=A128411(n,k)/(2^(k-1)+0^k/2).
Column k has g.f. if(k=0,1/(1+2*x),(1-2*x)*x^k/(1+2*x)^(2*k+1)).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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