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.

A178120 Coefficient array of orthogonal polynomials P(n,x)=(x-2n)*P(n-1,x)-(2n-3)*P(n-2,x), P(0,x)=1,P(1,x)=x-2.

Original entry on oeis.org

1, -2, 1, 7, -6, 1, -36, 40, -12, 1, 253, -326, 131, -20, 1, -2278, 3233, -1552, 324, -30, 1, 25059, -38140, 20678, -5260, 675, -42, 1, -325768, 523456, -310560, 90754, -14380, 1252, -56, 1, 4886521, -8205244, 5223602, -1694244, 312059, -33866, 2135, -72, 1
Offset: 0

Views

Author

Paul Barry, May 20 2010

Keywords

Comments

Inverse is A178121. First column is A112293 signed.

Examples

			Triangle begins
1,
-2, 1,
7, -6, 1,
-36, 40, -12, 1,
253, -326, 131, -20, 1,
-2278, 3233, -1552, 324, -30, 1,
25059, -38140, 20678, -5260, 675, -42, 1,
-325768, 523456, -310560, 90754, -14380, 1252, -56, 1,
4886521, -8205244, 5223602, -1694244, 312059, -33866, 2135, -72, 1
Production matrix of inverse is
2, 1,
1, 4, 1,
0, 3, 6, 1,
0, 0, 5, 8, 1,
0, 0, 0, 7, 10, 1,
0, 0, 0, 0, 9, 12, 1,
0, 0, 0, 0, 0, 11, 14, 1,
0, 0, 0, 0, 0, 0, 13, 16, 1,
0, 0, 0, 0, 0, 0, 0, 15, 18, 1

Programs

  • Maple
    A178120 := proc(n,k)
    if n = k then
        1;
    elif n = 1 and k = 0 then
        -2 ;
    elif k < 0 or k > n then
        0 ;
    else
        -2*n*procname(n-1,k)+procname(n-1,k-1)-(2*n-3)*procname(n-2,k) ;
    end if;
end proc: # R. J. Mathar, Dec 03 2014
  • Mathematica
    P[0, _] = 1;
P[1, x_] := x - 2;
P[n_, x_] := P[n, x] = (x-2n) P[n-1, x] - (2n-3) P[n-2, x];
T[n_] := Module[{x}, CoefficientList[P[n, x], x]];
Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Aug 06 2023 *)

A178119 Expansion of 1/(1-2x-x^2/(1-4x-3x^2/(1-6x-5x^2/(1-8x-7x^2/(1-...))))) (continued fraction).

Original entry on oeis.org

1, 2, 5, 16, 64, 308, 1727, 11008, 78244, 611060, 5184338, 47366320, 462782080, 4807659368, 52853722811, 612426360832, 7453621425532, 94997205901940, 1264555335831662, 17540102647480336, 252979919852470672
Offset: 0

Views

Author

Paul Barry, May 20 2010

Keywords

Comments

Hankel transform is A057863. First column of A178121.

Examples

			G.f. = 1 + 2*x + 5*x^2 + 16*x^3 + 64*x^4 + 308*x^5 + 1727*x^6 + 11008*x^7 + ...

Formula

G.f.: 1/(Q(0)-x) where Q(k) = 1 - x*(2*k+1)/( 1 - x/Q(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 22 2013

A185997 Inverse of coefficient array of orthogonal polynomials P(n,x)=(x-2n+2)*P(n-1,x)-(2n-3)*P(n-2,x), P(0,x)=1, P(1,x)=x-1.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 11, 7, 1, 16, 48, 44, 13, 1, 64, 244, 289, 129, 21, 1, 308, 1419, 2045, 1210, 306, 31, 1, 1727, 9281, 15649, 11447, 3937, 627, 43, 1, 11008, 67236, 129112, 111890, 48586, 10680, 1156, 57, 1, 78244, 532816, 1143134, 1140554, 596698, 168102, 25293, 1969, 73, 1, 611060, 4573278, 10808122, 12163344, 7427056, 2555941, 497215, 53954, 3154, 91, 1
Offset: 0

Views

Author

Paul Barry, Feb 08 2011

Keywords

Examples

			Triangle begins:
  1,
  1, 1,
  2, 3, 1,
  5, 11, 7, 1,
  16, 48, 44, 13, 1,
  64, 244, 289, 129, 21, 1,
  308, 1419, 2045, 1210, 306, 31, 1,
  1727, 9281, 15649, 11447, 3937, 627, 43, 1,
  11008, 67236, 129112, 111890, 48586, 10680, 1156, 57, 1,
  78244, 532816, 1143134, 1140554, 596698, 168102, 25293, 1969, 73, 1
Production matrix begins:
  1, 1,
  1, 2, 1,
  0, 3, 4, 1,
  0, 0, 5, 6, 1,
  0, 0, 0, 7, 8, 1,
  0, 0, 0, 0, 9, 10, 1,
  0, 0, 0, 0, 0, 11, 12, 1,
  0, 0, 0, 0, 0, 0, 13, 14, 1,
  0, 0, 0, 0, 0, 0, 0, 15, 16, 1

Links

  • E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.

Crossrefs

Cf. A178121.
Inverse is A185996. First column is A185998.
Showing 1-3 of 3 results.

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