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%I #49 Aug 10 2024 11:03:06
%S 1,1,1,2,5,10,18,35,73,151,306,623,1286,2668,5531,11477,23889,49852,
%T 104175,217936,456534,957609,2010839,4226417,8891022,18719637,
%U 39443860,83170162,175484915,370491775,782648333,1654197568,3498049053,7400639286,15664103420,33168342557,70260909811
%N Expansion of Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(2*k).
%C Compare to the g.f. of Narayana's cows sequence A000930:
%C Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k) * x^(2*k) = 1/(1-x-x^3).
%C Compare to the g.f. of Whitney numbers sequence A051286:
%C Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^k = 1/sqrt((1+x+x^2)*(1-3*x+x^2)).
%C ...
%C Lim_{n->infinity} a(n)/a(n+1) = t^2 = 0.465571231876768... (A088559) where t = ((sqrt(93)+9)/18)^(1/3) - ((sqrt(93)-9)/18)^(1/3) is the positive real root of 1 - x - x^3 = 0.
%H G. C. Greubel, <a href="/A246840/b246840.txt">Table of n, a(n) for n = 0..1000</a>
%H Hacène Belbachir and Abdelghani Mehdaoui, <a href="https://doi.org/10.2989/16073606.2020.1729269">Recurrence relation associated with the sums of square binomial coefficients</a>, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
%F G.f.: Sum_{n>=0} (2*n)!/(n!)^2 * x^(3*n) / (1 - x + x^3)^(2*n+1). - _Paul D. Hanna_, Oct 15 2014
%F G.f.: Sum_{n>=0} x^n * [Sum_{k>=0} C(n+k,k)^2 * x^(2*k)] * (1-x^2)^(2*n+1).
%F G.f.: Sum_{n>=0} x^(3*n) * [Sum_{k>=0} C(n+k,k)^2 * x^k].
%F G.f.: Sum_{n>=0} x^(3*n) * [Sum_{k=0..n} C(n,k)^2 * x^k] /(1-x)^(2n+1).
%F G.f.: exp( Sum_{n>=1} (x^n/n) * Sum_{k=0..n} C(2*n,2*k) * x^(2*k) ).
%F G.f.: exp( Sum_{n>=1} (x^n/n) * ((1+x)^(2*n) + (1-x)^(2*n))/2 ).
%F G.f.: 1 / sqrt((1 - x + 2*x^2 - x^3)*(1 - x - 2*x^2 - x^3)).
%F G.f.: 1 / sqrt((1 - x - x^3)^2 - 4*x^4).
%F a(n) = Sum_{k=0..floor(n/2)} C(n-2*k, k)^2.
%F n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + (2*n-3)*a(n-3) + 2*(n-2)*a(n-4) - (n-3)*a(n-6). - _Seiichi Manyama_, Aug 10 2024
%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 10*x^5 + 18*x^6 + 35*x^7 + ...
%e where, by definition,
%e A(x) = 1 + x*(1 + x^2) + x^2*(1 + 2^2*x^2 + x^4)
%e + x^3*(1 + 3^2*x^2 + 3^2*x^4 + x^6)
%e + x^4*(1 + 4^2*x^2 + 6^2*x^4 + 4^2*x^6 + x^8)
%e + x^5*(1 + 5^2*x^2 + 10^2*x^4 + 10^2*x^6 + 5^2*x^8 + x^10) + ...
%e which is also given by the series identity:
%e A(x) = 1/(1-x+x^3) + 2*x^3/(1-x+x^3)^3 + 6*x^6/(1-x+x^3)^5 + 20*x^9/(1-x+x^3)^7 + 70*x^12/(1-x+x^3)^9 + 252*x^15/(1-x+x^3)^11 + 924*x^18/(1-x+x^3)^13 + ...
%e The logarithm of the g.f. begins:
%e log(A(x)) = x*(1 + x^2) + x^2*(1 + 6*x^2 + x^4)/2
%e + x^3*(1 + 15*x^2 + 15*x^4 + x^6)/3
%e + x^4*(1 + 28*x^2 + 70*x^4 + 28*x^6 + x^8)/4
%e + x^5*(1 + 45*x^2 + 210*x^4 + 210*x^6 + 45*x^8 + x^10)/5 + ...
%e more explicitly,
%e log(A(x)) = x + x^2/2 + 4*x^3/3 + 13*x^4/4 + 26*x^5/5 + 46*x^6/6 + 99*x^7/7 + 229*x^8/8 + 499*x^9/9 + 1046*x^10/10 + 2223*x^11/11 + 4810*x^12/12 + ...
%e where the logarithmic derivative equals
%e A'(x)/A(x) = (1-x+3*x^2+4*x^3-3*x^5)/((1-x+2*x^2-x^3)*(1-x-2*x^2-x^3)).
%t CoefficientList[Series[1/Sqrt[(1 - x - x^3)^2 - 4*x^4], {x,0,50}], x] (* _G. C. Greubel_, Apr 27 2017 *)
%o (PARI) /* By definition: */
%o {a(n)=local(A=1);A=sum(m=0,n,x^m*sum(k=0,m,binomial(m,k)^2*x^(2*k)) +x*O(x^n));polcoeff(A,n)}
%o for(n=0,40,print1(a(n),", "))
%o (PARI) /* From closed formula: */
%o {a(n)=local(A=1);A= 1/sqrt((1 - x - x^3)^2 - 4*x^4 +x*O(x^n));polcoeff(A, n)}
%o for(n=0,40,print1(a(n),", "))
%o (PARI) /* From a series identity: */
%o {a(n)=local(A=1+x); A=sum(m=0, n, (2*m)!/(m!)^2 * x^(3*m) / (1 - x + x^3 +x*O(x^n))^(2*m+1)); polcoeff(A, n)}
%o for(n=0, 40, print1(a(n), ", "))
%o (PARI) /* From a binomial series identity: */
%o {a(n)=local(A=1+x); A=sum(m=0, n, x^m*(1-x^2)^(2*m+1)*sum(k=0, n, binomial(m+k, k)^2*x^(2*k)) +x*O(x^n)); polcoeff(A, n)}
%o for(n=0, 40, print1(a(n), ", "))
%o (PARI) /* From a binomial series identity: */
%o {a(n)=local(A=1+x); A=sum(m=0, n\3, x^(3*m)*sum(k=0, n-3*m, binomial(m+k, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)}
%o for(n=0, 40, print1(a(n), ", "))
%o (PARI) /* From a binomial series identity: */
%o {a(n)=local(A=1+x); A=sum(m=0, n\3, x^(3*m) * sum(k=0, m, binomial(m, k)^2*x^k) / (1-x +x*O(x^n))^(2*m+1) ); polcoeff(A, n)}
%o for(n=0, 40, print1(a(n), ", "))
%o (PARI) /* From exponential formula: */
%o {a(n)=local(A=1);A=exp(sum(m=1, n, sum(k=0, m, binomial(2*m, 2*k)*x^(2*k)) * x^m/m) +x*O(x^n));polcoeff(A, n)}
%o for(n=0,40,print1(a(n),", "))
%o (PARI) /* From exponential formula: */
%o {a(n)=local(A=1);A=exp(sum(m=1, n, ((1+x)^(2*m) + (1-x)^(2*m))/2 * x^m/m) +x*O(x^n));polcoeff(A, n)}
%o for(n=0,40,print1(a(n),", "))
%o (PARI) /* From formula for a(n): */
%o {a(n)=sum(k=0,n\2,binomial(n-2*k,k)^2)}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A088559, A246861, A181665, A246883, A246884, A248193.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Sep 04 2014
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