OFFSET
0,2
COMMENTS
Apply Riordan array (c(x)/sqrt(1-4*x), x*c(x)^2) to A131055.
Hankel transform appears to be (-1)^n*A085046(n).
Coefficients T(2*n+1,n) of triangle A103450. [Emanuele Munarini, Jun 01 2012, corrected by Werner Schulte, Nov 27 2021]
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
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).
Equals the Narayana transform (A001263) of integer squares. - Gary W. Adamson, Jul 29 2011
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
From Vaclav Kotesovec, Feb 13 2014: (Start)
G.f.: -1/(2*x) + (2*x-1)^2/(2*x*(1-4x)^(3/2)).
a(n) = (1 + 3*n + n^2) * C(2*n,n) / (n+1).
Recurrence: (n+1)*(n^2 + n - 1)*a(n) = 2*(2*n-1)*(n^2 + 3*n + 1)*a(n-1).
(End)
MATHEMATICA
Table[((1+3*n+n^2)*Binomial[2*n, n])/(n+1), {n, 0, 20}] (* Vaclav Kotesovec, Feb 13 2014 *)
CoefficientList[Series[-1/(2*x)+(2*x-1)^2/(2*x*(1-4x)^(3/2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)
a[n_] := (1 + 3 n + n^2) CatalanNumber[n];
Table[a[n], {n, 0, 21}] (* Peter Luschny, Nov 28 2021 *)
PROG
(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 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jun 14 2008
EXTENSIONS
Name of the sequence corrected by Vaclav Kotesovec, Feb 13 2014
STATUS
approved