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A245925
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G.f.: Sum_{n>=0} x^n*Sum_{k=0..n} (-1)^k * C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * x^j.
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16
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1, -3, 25, -243, 2601, -29403, 344569, -4141875, 50737129, -630663003, 7930793025, -100681224075, 1288236350025, -16592960274075, 214939203248025, -2797935722568243, 36578032462268649, -480000660000226875, 6320012816203363489, -83462977778600141643, 1105193229806740453201
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OFFSET
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0,2
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COMMENTS
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The g.f.s formed from a(2*n)^(1/2) and (-a(2*n+1)/3)^(1/2) are:
A245926: sqrt( (1-x + sqrt(1-14*x+x^2)) / (2*(1-14*x+x^2)) );
A245927: sqrt( (1-x - sqrt(1-14*x+x^2)) / (6*x*(1-14*x+x^2)) ).
Lim_{n->infinity} a(n+1)/a(n) = -(7 + 4*sqrt(3)).
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LINKS
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FORMULA
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G.f.: Sum_{n>=0} x^n / (1+x)^(2*n+1) * ( Sum_{k=0..n} C(n,k)^2*(-x)^k )^2.
G.f.: 1 / AGM(1-x^2, sqrt(1+14*x^2+x^4)), where AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) is the arithmetic-geometric mean.
a(n) = Sum_{k=0..n} Sum_{j=0..2*n-2*k} (-1)^(j+k) * C(2*n-k,j+k)^2 * C(j+k,k)^2.
D-finite with recurrence: n^2*(2*n-3)*a(n) = -(2*n-1)*(13*n^2 - 26*n + 10)*a(n-1) + (2*n-3)*(13*n^2 - 26*n + 10)*a(n-2) + (n-2)^2*(2*n-1)*a(n-3). - Vaclav Kotesovec, Aug 16 2014
a(n) ~ (-1)^n * (2+sqrt(3)) * (7+4*sqrt(3))^n / (4*Pi*n). - Vaclav Kotesovec, Aug 16 2014
a(n) = (-1)^n*Sum_{k=0..n} binomial(2*(n-k), n-k)*binomial(2*n-k, k)^2. - Peter Luschny, Aug 17 2014
a(n) = (-1)^n*binomial(2*n,n)*hyper4F3([-n,-n,-n,-n+1/2],[1,-2*n,-2*n], 4). - Peter Luschny, Aug 17 2014
a(n) = Sum_{k=0..n} (-1)^k * C(2*k, k)^2 * C(n+k, n-k). - Paul D. Hanna, Aug 17 2014
a(n) = (-1)^n*hypergeom([-n, -n, n + 1, n + 1], [1/2, 1, 1], 1/4). - Peter Luschny, Mar 14 2018
a(n) = Legendre_P(n, sqrt(-3))^2. - Peter Bala, Dec 22 2020
G.f.: Sum_{n >= 0} (-1)^n*binomial(2*n,n)^2*x^n/(1-x)^(2*n+1). - Peter Bala, Feb 07 2022
a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(2*n-k,k)*binomial(2*n-2*k,n-k)^2.
a(n) = (-1)^n * Sum_{k = 0..n} binomial(2*k,k)*binomial(n+k,n-k)^2.
a(n) = (-1)^n*binomial(2*n,n)^2*hypergeom([-n,-n,-n,], [-2*n,-n+1/2], 1/4). (End)
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EXAMPLE
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G.f.: A(x) = 1 - 3*x^2 + 25*x^4 - 243*x^6 + 2601*x^8 - 29403*x^10 + ...
where the g.f. is given by the binomial series:
A(x) = 1 + x*(1 - (1+x)) + x^2*(1 - 2^2*(1+x) + (1+2^2*x+x^2))
+ x^3*(1 - 3^2*(1+x) + 3^2*(1+2^2*x+x^2) - (1+3^2*x+3^2*x^2+x^3))
+ x^4*(1 - 4^2*(1+x) + 6^2*(1+2^2*x+x^2) - 4^2*(1+3^2*x+3^2*x^2+x^3) + (1+4^2*x+6^2*x^2+4^2*x^3+x^4))
+ x^5*(1 - 5^2*(1+x) + 10^2*(1+2^2*x+x^2) - 10^2*(1+3^2*x+3^2*x^2+x^3) + 5^2*(1+4^2*x+6^2*x^2+4^2*x^3+x^4) - (1+5^2*x+10^2*x^2+10^2*x^3+5^2*x^4+x^5))
+ x^6*(1 - 6^2*(1+x) + 15^2*(1+2^2*x+x^2) - 20^2*(1+3^2*x+3^2*x^2+x^3) + 15^2*(1+4^2*x+6^2*x^2+4^2*x^3+x^4) - 6^2*(1+5^2*x+10^2*x^2+10^2*x^3+5^2*x^4+x^5) + (1+6^2*x+15^2*x^2+20^2*x^3+15^2*x^4+6^2*x^5+x^6)) + ...
in which the coefficients of odd powers of x vanish.
We can also express the g.f. by the binomial series identity:
A(x) = 1/(1+x) + x/(1+x)^3*(1-x)^2 + x^2/(1+x)^5*(1 - 2^2*x + x^2)^2
+ x^3/(1+x)^7*(1 - 3^2*x + 3^2*x^2 - x^3)^2
+ x^4/(1+x)^9*(1 - 4^2*x + 6^2*x^2 - 4^2*x^3 + x^4)^2
+ x^5/(1+x)^11*(1 - 5^2*x + 10^2*x^2 - 10^2*x^3 + 5^2*x^4 - x^5)^2
+ x^6/(1+x)^13*(1 - 6^2*x + 15^2*x^2 - 20^2*x^3 + 15^2*x^4 - 6^2*x^5 + x^6)^2 + ...
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MAPLE
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A245925 := n -> (-1)^n*add(binomial(2*(n-k), n-k)*binomial(2*n-k, k)^2, k=0..n); seq(A245925(n), n=0..20); # Peter Luschny, Aug 17 2014
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MATHEMATICA
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Table[Sum[Sum[(-1)^(j+k) * Binomial[2*n - k, j + k]^2 * Binomial[j + k, k]^2, {j, 0, 2*n - 2*k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 16 2014 after Paul D. Hanna *)
a[n_] := (-1)^n*HypergeometricPFQ[{-n, -n, n + 1, n + 1}, {1/2, 1, 1}, 1/4];
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PROG
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(PARI) /* By definition: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, (-1)^k*binomial(m, k)^2*sum(j=0, k, binomial(k, j)^2*x^j)+x*O(x^n))), n)}
for(n=0, 20, print1(a(2*n), ", "))
(PARI) /* From alternate g.f.: */
{a(n)=local(A=1); A=sum(m=0, n, x^m/(1+x)^(2*m+1)*sum(k=0, m, binomial(m, k)^2*(-x)^k)^2+x*O(x^n)); polcoeff(A, n)}
for(n=0, 20, print1(a(2*n), ", "))
(PARI) /* From formula for a(n); printing only nonzero terms: */
{a(n)=sum(k=0, n\2, sum(j=0, n-2*k, (-1)^(j+k)*binomial(n-k, j+k)^2*binomial(j+k, k)^2))}
for(n=0, 20, print1(a(2*n), ", "))
(PARI) /* From formula for a(n) (nonzero terms): */
{a(n)=sum(k=0, n, sum(j=0, 2*n-2*k, (-1)^(j+k)*binomial(2*n-k, j+k)^2*binomial(j+k, k)^2))}
for(n=0, 20, print1(a(n), ", "))
{a(n) = (-1)^n * sum(k=0, n, binomial(2*k, k) * binomial(n+k, n-k)^2)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Simpler formula for a(n): */
{a(n) = sum(k=0, n, (-1)^k * binomial(2*k, k)^2 * binomial(n+k, n-k) )}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Using AGM: */
{a(n)=polcoeff( 1 / agm(1-x, sqrt(1+14*x+x^2 +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
(Sage)
A245925 = lambda n: (-1)^n*sum(binomial(2*(n-k), n-k)*binomial(2*n-k, k)^2 for k in (0..n))
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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