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.

A238731 Riordan array ((1-2*x)/(1-3*x+x^2), x/(1-3*x+x^2)).

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 5, 13, 7, 1, 13, 40, 33, 10, 1, 34, 120, 132, 62, 13, 1, 89, 354, 483, 308, 100, 16, 1, 233, 1031, 1671, 1345, 595, 147, 19, 1, 610, 2972, 5561, 5398, 3030, 1020, 203, 22, 1, 1597, 8495, 17984, 20410, 13893, 5943, 1610, 268, 25, 1, 4181
Offset: 0

Views

Author

Philippe Deléham, Mar 03 2014

Keywords

Comments

Unsigned version of A124037 and A126126.
Subtriangle of the triangle given by (0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Row sums are A001075(n).
Diagonal sums are A133494(n).
Sum_{k=0..n} T(n,k)*x^k = A001519(n), A001075(n), A002320(n), A038723(n), A033889(n) for x = 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Mar 05 2014

Examples

			Triangle begins:
1;
1, 1;
2, 4, 1;
5, 13, 7, 1;
13, 40, 33, 10, 1;
34, 120, 132, 62, 13, 1;
89, 354, 483, 308, 100, 16, 1;
233, 1031, 1671, 1345, 595, 147, 19, 1;...
Triangle (0, 1, 1, 1, 0, 0, 0, ...) DELTA (1, 0, 2, -2, 0, 0, ...) begins:
1;
0, 1;
0, 1, 1;
0, 2, 4, 1;
0, 5, 13, 7, 1;
0, 13, 40, 33, 10, 1;
0, 34, 120, 132, 62, 13, 1;
0, 89, 354, 483, 308, 100, 16, 1;
0, 233, 1031, 1671, 1345, 595, 147, 19, 1;...

Crossrefs

Cf. A001519, A000012, A016777, A062708.
Cf. A001906, A124037, A126126.

Programs

  • Mathematica
    (* The function RiordanArray is defined in A256893. *)
RiordanArray[(1-2#)/(1-3#+#^2)&, x/(1-3#+#^2)&, 10] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Formula

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
G.f.: (1-2*x)/(1-(y+3)*x+x^2). - Philippe Deléham, Mar 05 2014

A126126 Triangle read by rows: matrix inverse of A110877.

Original entry on oeis.org

1, -1, 1, 2, -4, 1, -5, 13, -7, 1, 13, -40, 33, -10, 1, -34, 120, -132, 62, -13, 1, 89, -354, 483, -308, 100, -16, 1, -233, 1031, -1671, 1345, -595, 147, -19, 1, 610, -2972, 5561, -5398, 3030, -1020, 203, -22, 1, -1597, 8495, -17984, 20410, -13893, 5943, -1610, 268, -25, 1, 4181, -24110, 56886, -73816, 59059, -30702, 10570, -2392, 342, -28, 1
Offset: 0

Views

Author

Gary W. Adamson, Dec 17 2006

Keywords

Comments

A110877 can be generated from M^n * [1,0,0,0...] where M = an infinite tridiagonal matrix with 1's in the super and subdiagonals and (1,3,3,3...) in the main diagonal.
Up to signs the same as A124037. - R. J. Mathar, Sep 06 2013
Riordan array ((1+2*x)/(1+3*x+x^2), x/(1+3*x+x^2)). - Philippe Deléham, Mar 04 2014

Examples

			First few rows of the triangle are:
1;
-1, 1;
2, -4, 1;
-5, 13, -7, 1;
13, -40, 33, -10, 1;
-34, 120, -132, 62, -13, 1
...

Crossrefs

Cf. A110877.

Formula

Sum_{j=k..n} T(n,j)*A110877(j,k) = delta(n,k).

Extensions

Corrected by R. J. Mathar, Sep 06 2013

A238846 Expansion of (1-2*x)/(1-3*x+x^2)^2.

Original entry on oeis.org

1, 4, 13, 40, 120, 354, 1031, 2972, 8495, 24110, 68016, 190884, 533293, 1484020, 4115185, 11375764, 31358376, 86223942, 236540915, 647556620, 1769374931, 4826148314, 13142564448, 35736448200, 97037995225, 263156279524, 712795854421, 1928547574912, 5212430732760
Offset: 0

Views

Author

Philippe Deléham, Mar 05 2014

Keywords

Comments

Convolution of 1, 1, 2, 5, 13, ... (A001519(n)) with 1, 3, 8, 21, 55, ... (A001906(n+1)).

Examples

			a(0) = 1*1 = 1;
a(1) = 1*3 + 1*1 = 4;
a(2) = 1*8 + 1*3 + 2*1 = 13;
a(3) = 1*21 + 1*8 + 2*3 + 5*1 = 40;
a(4) = 1*55 + 1*21 + 2*8 + 5*3 + 13*1 = 120; etc. (from first recurrence formula).
a(0) = 3*0 - 0 + 1 = 1;
a(1) = 3*1 - 0 + 1 = 4;
a(2) = 3*4 - 1 + 2 = 13;
a(3) = 3*13 - 4 + 5 = 40;
a(4) = 3*40 - 13 + 13 = 120; etc (from second recurrence formula).
G.f. = 1 + 4*x + 13*x^2 + 40*x^3 + 120*x^4 + 354*x^5 + 1031*x^6 + ... - _Michael Somos_, Nov 23 2021

Links

Crossrefs

Cf. A000045, A001519, A001906.
Cf. A001870, A001871, A197649.

Programs

  • Mathematica
    LinearRecurrence[{6, -11, 6, -1}, {1, 4, 13, 40}, 30] (* Bruno Berselli, Mar 06 2014 *)
a[ n_] := If[n < 0, SeriesCoefficient[ x^3*(2 - x)/(1 - 3*x + x^2)^2, {x, 0, -n}], SeriesCoefficient[ (1 - 2*x)/(1 - 3*x + x^2)^2, {x, 0, n}]]; (* Michael Somos, Nov 23 2021 *)
  • PARI
    {a(n) = if(n<0, polcoeff( x^3*(2-x)/(1-3*x+x^2)^2 + x*O(x^-n), -n), polcoeff( (1-2*x)/(1-3*x+x^2)^2 + x*O(x^n), n))}; /* Michael Somos, Nov 23 2021 */

Formula

a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4) for n>3, a(0)=1, a(1)=4, a(2)=13, a(3)=40.
a(n) = 3*a(n-1) - a(n-2) + A001519(n) for n>1, a(0)=1, a(1)=4.
a(n) = A238731(n+1, 1).
a(n) = -A124037(n+1, 1).
a(n) = (-1)^n*A126126(n+1, 1).
a(n) = ( (3 + sqrt(5))^(1+n)*(8 - (1 - sqrt(5))*(13 + 5*n)) + (3 - sqrt(5))^(1+n)*(8 - (1 + sqrt(5))*(13 + 5*n)) ) / (25*2^(2+n)). - Bruno Berselli, Mar 06 2014
From Philippe Deléham, Mar 06 2014: (Start)
a(n) = 2*A001870(n) - A001871(n).
a(n) = A197649(n+1) - 3*A001871(n-1).
a(n) = A001871(n) - 2*A001871(n-1). (End)
0 = 2 + a(n)*(a(n+1) - a(n+3)) + a(n+1)*(-6*a(n+1) + 12*a(n+2)) + a(n+2)*(-6*a(n+2) + a(n+3)) for all n in Z. - Michael Somos, Nov 23 2021
E.g.f.: exp(3*x/2)*(5*(5 + 4*x)*cosh(sqrt(5)*x/2) + sqrt(5)*(17 + 10*x)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Mar 04 2025
Showing 1-3 of 3 results.

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