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 1-5 of 5 results.

A174016 Row sums of number triangle A174014.

Original entry on oeis.org

1, 2, 6, 16, 40, 92, 192, 352, 528, 512, -192, -2128, -3840, 5888, 69056, 299264, 917760, 2125184, 3258368, -117760, -23297536, -103321600, -295843840, -577744128, -416948224, 2490589184, 14821469184, 50063536128
Offset: 0

Views

Author

Paul Barry, Mar 05 2010

Keywords

Comments

Hankel transform is 1, 2, -8, -32, -256, 4096, ... (signed version of A134751).

Formula

G.f.: (sqrt(1-4x+4x^2+8x^3)+4x-1)/(2x(1-2x));
g.f.: 1/(1-2x/(1-x/(1+x/(1-2x/(1-x/(1+x/(1-2x/(1-... (continued fraction).
a(n) = Sum_{k=0..n} A198379(n,k)*2^k. - Philippe Deléham, Oct 29 2011
Conjecture: (n+1)*a(n) - 6*n*a(n-1) + 12*(n-1)*a(n-2) - 12*a(n-3) + 8*(7-2*n)*a(n-4) = 0. - R. J. Mathar, Nov 13 2012

Extensions

First formula corrected by Philippe Deléham, Feb 16 2012

A198379 Triangle T(n,k), read by rows, given by (0,1,-1,0,1,-1,0,1,-1,0,1,-1,...) DELTA (1,0,0,1,0,0,1,0,0,1,0,0,1,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 0, 3, 1, 0, 0, -1, 0, 4, 1, 0, 0, 0, -4, 0, 5, 1, 0, 0, 0, 0, -10, 0, 6, 1, 0, 0, 0, 2, 0, -20, 0, 7, 1, 0, 0, 0, 0, 12, 0, -35, 0, 8, 1, 0, 0, 0, 0, 0, 42, 0, -56, 0, 9, 1
Offset: 0

Views

Author

Philippe Deléham, Oct 28 2011

Keywords

Comments

Equal to A174014*A130595 as infinite lower triangular matrices.

Examples

			Triangle begins :
1
0, 1
0, 1, 1
0, 0, 2, 1
0, 0, 0, 3, 1
0, 0, -1, 0, 4, 1
0, 0, 0, -4, 0, 5, 1
0, 0, 0, 0, -10, 0, 6, 1

Crossrefs

Cf. A174014, A174015, A174016

Formula

Sum_k>=0 T(n,k)= A174015(n).
Sum_k>=0 T(n,k)*2^k = A174016(n).
Sum_ {0<=k<=n} T(n,k)*(-1)^(n-k) = A168505(n).

A168505 Expansion of 1/(1-x/(1+x/(1-x/(1-x/(1+x/(1-x/(1-x/(1+x/(1-... (continued fraction).

Original entry on oeis.org

1, 1, 0, -1, -2, -2, 0, 5, 12, 16, 6, -32, -102, -170, -130, 199, 966, 1978, 2192, -650, -9292, -23624, -33760, -12138, 84440, 280852, 493932, 397668, -639676, -3248464, -6947460, -8068587, 2165980, 35591960, 94129446, 139864828, 56393482, -352505722
Offset: 0

Views

Author

Paul Barry, Nov 27 2009

Keywords

Comments

Hankel transform is A131561(n+1). First column of array whose production matrix begins
1, 1;
-1, 0, 1;
0, 1, 0, 1;
0, 0, -1, 2, 1;
0, 0, 0, -1, 0, 1;
0, 0, 0, 0, 1, 0, 1;
0, 0, 0, 0, 0, -1, 2, 1;

Examples

			G.f. = 1 + x - x^3 - 2*x^4 - 2*x^5 + 5*x^7 + 12*x^8 + 16*x^9 + 6*x^10 + ...

Programs

  • PARI
    {a(n) = if( n<0, 0, polcoeff( (1 + x - sqrt(1 - 2*x + x^2 + 4*x^3 + x^2 * O(x^n))) / (2*x*(1 - x)), n))}; /* Michael Somos, Jan 20 2017 */

Formula

G.f.: 1/(1-x+x^2/(1-x^2/(1+x^2/(1-2x+x^2/(1-x^2/(1+x^2/(1-2x+x^2/(1-x^2/(1+... (continued fraction, defined by the sequences (1,0,0,2,0,0,2,0,0,2,0,...) and (-1,1,-1,-1,1,-1,...));
g.f.: (1+x-sqrt(1-2x+x^2+4x^3))/(2x(1-x)).
a(n) = Sum_{k=0..n} A198379(n,k)*(-1)^(n-k). - Philippe Deléham, Oct 29 2011
a(n) = (-1)^n*Sum_{k=0..n} A174014(n,k)*(-2)^k. - Philippe Deléham, Feb 16 2012
G.f.: (1+x)/(G(0)+x), where G(k) = 1 - x + x^3/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 29 2013
Conjecture: (n+1)*a(n) - 3*n*a(n-1) + 3*(n-1)*a(n-2) + 3*(n-4)*a(n-3) + 2*(-2*n+7)*a(n-4) = 0. - R. J. Mathar, Feb 10 2015
G.f. A(x) satisfies (A(x) - 1) / A(x)^2 = (x - x^2) / (1 + x). - Michael Somos, Jan 20 2017
0 = a(n)*(+16*a(n+1) - 6*a(n+2) - 42*a(n+3) + 54*a(n+4) - 22*a(n+5)) + a(n+1)*(-18*a(n+1) + 27*a(n+2) + 6*a(n+3) - 31*a(n+4) + 18*a(n+5))+ a(n+2)*(-18*a(n+2) + 36*a(n+3) - 30*a(n+4) + 9*a(n+5)) + a(n+3)*(+6*a(n+4) - 6*a(n+5)) + a(n+4)*(+a(n+5)) if n >= 0. - Michael Somos, Jan 20 2017

A174013 Sequence whose Hankel transform is a (1,1) Somos-4 sequence.

Original entry on oeis.org

1, 1, 3, 6, 12, 22, 37, 56, 73, 75, 44, -21, -39, 297, 1751, 5749, 14104, 27136, 38163, 22135, -80421, -369611, -934754, -1637758, -1559395, 2019629, 14766699, 44732254, 94865112, 138114302, 61077521
Offset: 0

Views

Author

Paul Barry, Mar 05 2010

Keywords

Comments

Continued fraction form of g.f. A(x) given by A(x) = 1/(1-x*(1+x)/(1-x/(1+x*A(x)))).
Hankel transform is A174017.
Diagonal sums of the Deleham array [1,1,-1,1,1,-1,1,...] Delta [1,0,0,1,0,0,1,0,0,1,0,...], or A174014.

Examples

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 12*x^4 + 22*x^5 + 37*x^6 + ... - _Michael Somos_, Jul 11 2024

Crossrefs

Cf. A174014, A174017.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[2*(1-x)/(1-3*x-x^2 + Sqrt[1-2*x-x^2+6*x^3+5*x^4]), {x, 0, n}]; (* Michael Somos, Jul 11 2024 *)
  • PARI
    {a(n) = polcoeff(2*(1-x)/(1-3*x-x^2 + sqrt(1-2*x-x^2+6*x^3+5*x^4 + x*O(x^n))), n)}; /* Michael Somos, Jul 11 2024 */

Formula

G.f.: -(1-3*x-x^2-sqrt(1-2*x-x^2+6*x^3+5*x^4))/(2*x*(1-x-x^2)).
G.f.: 1/(1-x*(1+x)/(1-x/(1+x/(1-x*(1+x)/(1-x/(1+x/(1-... (continued fraction).
Conjecture: (n+1)*a(n) +n*a(n-1) +12*(-n+1)*a(n-2) +3*(3*n-8)*a(n-3) +6*(6*n-23)*a(n-4) +11*(-n+2)*a(n-5) +(-49*n+253)*a(n-6) +20*(-n+6)*a(n-7)=0. - R. J. Mathar, Jan 12 2013

A174015 A generalized Catalan number sequence.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 2, -3, -10, -14, -4, 34, 104, 172, 132, -197, -964, -1976, -2190, 652, 9294, 23626, 33762, 12140, -84438, -280850, -493930, -397666, 639678, 3248466, 6947462, 8068589, -2165978, -35591958, -94129444, -139864826, -56393480, 352505724
Offset: 0

Views

Author

Paul Barry, Mar 05 2010

Keywords

Comments

Hankel transform is A130151(n+1). First column of A174014.

Formula

G.f.: (sqrt(1-2x+x^2+4x^3)+3x-1)/(2x(1-x));
G.f.: 1/(1-x/(1-x/(1+x/(1-x/(1-x/(1+x/(1-... (continued fraction).
a(n) = Sum_{k, 0<=k<=n} A091866(n,k)*(-1)^(n-k) = Sum_{k, 0<=k<=n} A198379(n,k). - Philippe Deléham, Nov 27 2011
Conjecture: (n+1)*a(n) -3*n*a(n-1) +3*(n-1)*a(n-2) +3*(n-4)*a(n-3) +2*(7-2*n)*a(n-4)=0. R. J. Mathar, Nov 13 2012

Extensions

More terms from Philippe Deléham, Oct 27 2011
Showing 1-5 of 5 results.

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