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A141222 Expansion of -1/(2*x) + (2*x-1)^2/(2*x*(1-4x)^(3/2)).

%I A141222 #27 Nov 28 2021 08:12:25
%S A141222 1,5,22,95,406,1722,7260,30459,127270,529958,2200276,9111830,37650172,
%T A141222 155266100,639191160,2627302995,10784089350,44208873390,181025067300,
%U A141222 740483276610,3026059513620,12355464845100
%N A141222 Expansion of -1/(2*x) + (2*x-1)^2/(2*x*(1-4x)^(3/2)).
%C A141222 Apply Riordan array (1/sqrt(1-4x), xc(x)) to A131056, c(x) the g.f. of A000108.
%C A141222 Apply Riordan array (c(x)/sqrt(1-4*x), x*c(x)^2) to A131055.
%C A141222 Hankel transform appears to be (-1)^n*A085046(n).
%C A141222 Coefficients T(2*n+1,n) of triangle A103450. [_Emanuele Munarini_, Jun 01 2012, corrected by _Werner Schulte_, Nov 27 2021]
%H A141222 Vincenzo Librandi, <a href="/A141222/b141222.txt">Table of n, a(n) for n = 0..200</a>
%F A141222 a(n) = Sum_{k=0..n} (1 + (k+1)*2^(k-1) - 0^k/2)*C(2n-k,n-k); a(n) = Sum_{k=0..n} C(2n,k)*C(n+1,2n-k).
%F A141222 Equals the Narayana transform (A001263) of integer squares. - _Gary W. Adamson_, Jul 29 2011
%F A141222 Conjecture: (n+1)*a(n) + 2*(-3*n-1)*a(n-1) + 4*(2*n-3)*a(n-2) = 0. - _R. J. Mathar_, Nov 24 2012
%F A141222 From _Vaclav Kotesovec_, Feb 13 2014: (Start)
%F A141222 G.f.: -1/(2*x) + (2*x-1)^2/(2*x*(1-4x)^(3/2)).
%F A141222 a(n) = (1 + 3*n + n^2) * C(2*n,n) / (n+1).
%F A141222 Recurrence: (n+1)*(n^2 + n - 1)*a(n) = 2*(2*n-1)*(n^2 + 3*n + 1)*a(n-1).
%F A141222 (End)
%t A141222 Table[((1+3*n+n^2)*Binomial[2*n, n])/(n+1),{n,0,20}] (* _Vaclav Kotesovec_, Feb 13 2014 *)
%t A141222 CoefficientList[Series[-1/(2*x)+(2*x-1)^2/(2*x*(1-4x)^(3/2)),{x,0,20}],x] (* _Vaclav Kotesovec_, Feb 13 2014 *)
%t A141222 a[n_] := (1 + 3 n + n^2) CatalanNumber[n];
%t A141222 Table[a[n], {n, 0, 21}] (* _Peter Luschny_, Nov 28 2021 *)
%o A141222 (Maxima) a(n):=sum(binomial(2*n,k)*binomial(n+1,2*n-k),k,0,n); makelist(a(n),n,0,40); /* _Emanuele Munarini_, Jun 01 2012 */
%K A141222 easy,nonn
%O A141222 0,2
%A A141222 _Paul Barry_, Jun 14 2008
%E A141222 Name of the sequence corrected by _Vaclav Kotesovec_, Feb 13 2014