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A141222
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Expansion of -1/(2*x) + (2*x-1)^2/(2*x*(1-4x)^(3/2)).
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6
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1, 5, 22, 95, 406, 1722, 7260, 30459, 127270, 529958, 2200276, 9111830, 37650172, 155266100, 639191160, 2627302995, 10784089350, 44208873390, 181025067300, 740483276610, 3026059513620, 12355464845100
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OFFSET
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0,2
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COMMENTS
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Apply Riordan array (1/sqrt(1-4x), xc(x)) to A131056, c(x) the g.f. of A000108.
Apply Riordan array (c(x)/sqrt(1-4*x), x*c(x)^2) to A131055.
Hankel transform appears to be (-1)^n*A085046(n).
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LINKS
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FORMULA
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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).
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
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)
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MATHEMATICA
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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];
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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