A185965 -id:A185965 - cp's OEIS frontend

A185962 Riordan array ((1-x)^2/(1-x+x^2), x(1-x)^2/(1-x+x^2)).

1, -1, 1, -1, -2, 1, 0, -1, -3, 1, 1, 2, 0, -4, 1, 1, 3, 5, 2, -5, 1, 0, 0, 3, 8, 5, -6, 1, -1, -4, -6, -1, 10, 9, -7, 1, -1, -4, -10, -16, -10, 10, 14, -8, 1, 0, 1, 0, -10, -26, -24, 7, 20, -9, 1, 1, 6, 15, 20, 5, -30, -42, 0, 27, -10, 1

Offset: 0

Paul Barry, Feb 07 2011

Riordan array (g(x),xg(x)) where g(x)=(1-x)(1-x^2)(1-x^3)/(1-x^6).
Inverse is A185967. Row sums are A185963.
Diagonal sums are A185964. Central coefficients are A185965.
Subtriangle of the triangle given by (0, -1, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 19 2012

			Triangle begins:
   1;
  -1,  1;
  -1, -2,   1;
   0, -1,  -3,   1;
   1,  2,   0,  -4,   1;
   1,  3,   5,   2,  -5,   1;
   0,  0,   3,   8,   5,  -6,   1;
  -1, -4,  -6,  -1,  10,   9,  -7,  1;
  -1, -4, -10, -16, -10,  10,  14, -8,  1;
   0,  1,   0, -10, -26, -24,   7, 20, -9,   1;
   1,  6,  15,  20,   5, -30, -42,  0, 27, -10, 1;
  ...
From _Philippe Deléham_, Mar 19 2012: (Start)
(0, -1, 2, -1/2, 1/2, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
  1;
  0,  1;
  0, -1,  1;
  0, -1, -2,  1;
  0,  0, -1, -3,  1;
  0,  1,  2,  0, -4,  1;
  0,  1,  3,  5,  2, -5,  1;
  ... (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A195339, A195350.

CoefficientList[CoefficientList[Series[1/(1 - y*x + x/(1 - x)^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Jul 23 2017 *)

T(n,k) = Sum_{i=0..(2*k+2)} C(2*k+2,i)*Sum_{j=0..(n-k-i)} C(k+j,j)*C(j,n-k-i-j)*(-1)^(n-k-j).
G.f.: 1/(1-y*x+x/(1-x)^2). - Philippe Deléham, Feb 07 2012
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1) + T(n-3,k-1), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = T(2,0) = -1, T(2,1) = -2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham , Nov 11 2013

A185966 Series reversion of A028310.

Original entry on oeis.org

1, -1, 0, 2, -2, -5, 14, 5, -72, 68, 278, -726, -520, 4691, -3514, -21758, 50374, 56185, -374566, 194596, 1962618, -3956504, -6258320, 33057877, -8974630, -190822072, 330170022, 710487590, -3088268200, 18008739, 19398384974, -28292606291, -81631282280, 298546543220, 84094857302, -2028216574806, 2428288153424, 9450205225145

Offset: 0

Paul Barry, Feb 07 2011

			1 - x + 2*x^3 - 2*x^4 - 5*x^5 + 14*x^6 + 5*x^7 - 72*x^8 + 68*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A028130, A185962, A185965.

CoefficientList[1/x*InverseSeries[Series[x*(1-x+x^2) /(1-x)^2, {x, 0, 20}], x],x] (* Vaclav Kotesovec, Jan 22 2014 *)

{a(n) = if( n<0, 0, polcoeff( serreverse( x * ((1 - x + x^2) / (1 - x)^2 + x * O(x^n))) / x, n))} /* Michael Somos, Apr 05 2012 */

{a(n) = local(B); if( n<0, 0, B = O(x); for( k=0, n, B = (1 - B) * (x + B * (B - x))); polcoeff( B / x, n))} /* Michael Somos, Apr 05 2012 */

a(n) = A185962(2*n,n)/(n+1) = A185965(n)/(n+1).
Given g.f. A(x) then B(x) = x * A(x) satisfies B(x) = (1 - B(x)) * (x + B(x) * (B(x) - x)). - Michael Somos, Apr 05 2012
Conjecture: 6*n*(n+1)*a(n) -n*(n-14)*a(n-1) +2*n*(14*n-19)*a(n-2) -4*(n-2)*(17*n-48)*a(n-3) +6*(2*n-5)*(n-4)*a(n-4)=0. - R. J. Mathar, Nov 15 2012
Recurrence (of order 3): 3*n*(n+1)*(19*n-27)*a(n) = -2*n*(38*n^2 - 73*n + 9)*a(n-1) - 20*(19*n^3 - 65*n^2 + 66*n - 18)*a(n-2) + 2*(n-3)*(2*n-3)*(19*n-8)*a(n-3). - Vaclav Kotesovec, Jan 22 2014
Lim sup n->infinity |a(n)|^(1/n) = sqrt(20/9 + 1/27*(272376 - 12312 * sqrt(57))^(1/3) + 2/9*(1261 + 57 * sqrt(57))^(1/3)) = 2.637962913244886521522... - Vaclav Kotesovec, Jan 22 2014

cp's OEIS Frontend

A185962 Riordan array ((1-x)^2/(1-x+x^2), x(1-x)^2/(1-x+x^2)).

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