A198379 -id:A198379 - cp's OEIS frontend

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

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

Paul Barry, Nov 27 2009

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;

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

{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 */

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

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

Paul Barry, Mar 05 2010

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

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

More terms from Philippe Deléham, Oct 27 2011

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

Paul Barry, Mar 05 2010

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

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

First formula corrected by Philippe Deléham, Feb 16 2012

A196182 Triangle T(n,k), read by rows, given by (0,1,1,0,1,1,0,1,1,0,1,1,0,...) 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, 2, 2, 1, 0, 4, 6, 3, 1, 0, 8, 17, 12, 4, 1, 0, 16, 46, 44, 20, 5, 1, 0, 32, 120, 150, 90, 30, 6, 1, 0, 64, 304, 482, 370, 160, 42, 7, 1, 0, 128, 752, 1476, 1412, 770, 259, 56, 8, 1, 0, 256, 1824, 4344, 5068, 3402, 1428, 392, 72, 9, 1

Offset: 0

Philippe Deléham, Oct 28 2011

Row sums are A000108 ; diagonal sums are A005043, antidiagonal sums are A007477.

			Triangle begins :
1
0, 1
0, 1, 1
0, 2, 2, 1
0, 4, 6, 3, 1
0, 8, 17, 12, 4, 1
0, 16, 46, 44, 20, 5, 1

Cf. A084938, A175136, A198379

cp's OEIS Frontend

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

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A174015 A generalized Catalan number sequence.

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A174016 Row sums of number triangle A174014.

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Extensions

A196182 Triangle T(n,k), read by rows, given by (0,1,1,0,1,1,0,1,1,0,1,1,0,...) DELTA (1,0,0,1,0,0,1,0,0,1,0,0,1,...) where DELTA is the operator defined in A084938.

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