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A246883 Expansion of Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(3*k).

%I A246883 #34 Apr 29 2025 08:53:41
%S A246883 1,1,1,1,2,5,10,17,27,46,86,165,308,558,1006,1841,3421,6383,11863,
%T A246883 21966,40697,75662,141099,263429,491778,918104,1715259,3208078,
%U A246883 6005818,11250198,21082487,39524241,74135187,139128897,261228200,490682127,922015964,1733127107,3258939997,6130162494,11534742080
%N A246883 Expansion of Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(3*k).
%C A246883 Limit a(n)/a(n+1) = t^2 = 0.524888598656404... (A072223) where t is the positive real root of 1 - x - x^4 = 0.
%C A246883 Diagonal of the rational function 1 / ((1-x)*(1-y) - (x*y)^4). - _Seiichi Manyama_, Apr 29 2025
%H A246883 Michael De Vlieger, <a href="/A246883/b246883.txt">Table of n, a(n) for n = 0..1000</a>
%H A246883 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 A246883 G.f.: Sum_{n>=0} (2*n)!/(n!)^2 * x^(4*n) / (1 - x + x^4)^(2*n+1). - _Paul D. Hanna_, Oct 15 2014
%F A246883 G.f.: Sum_{n>=0} x^n * [Sum_{k>=0} C(n+k,k)^2 * x^(3*k)] * (1-x^3)^(2*n+1).
%F A246883 G.f.: Sum_{n>=0} x^(4*n) * [Sum_{k>=0} C(n+k,k)^2 * x^k].
%F A246883 G.f.: Sum_{n>=0} x^(4*n) * [Sum_{k=0..n} C(n,k)^2 * x^k] /(1-x)^(2n+1).
%F A246883 G.f.: exp( Sum_{n>=1} (x^n/n) * Sum_{k=0..n} C(2*n,2*k) * x^(3*k) ).
%F A246883 G.f.: 1 / sqrt((1 - x + 2*x^2 + x^4)*(1 - x - 2*x^2 + x^4)).
%F A246883 G.f.: 1 / sqrt((1 - x + x^4)^2 - 4*x^4).
%F A246883 G.f.: 1 / sqrt((1 - x - x^4)^2 - 4*x^5).
%F A246883 a(n) = Sum_{k=0..[n/3]} C(n-3*k, k)^2.
%F A246883 n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + 2*(n-2)*a(n-4) + (2*n-5)*a(n-5) - (n-4)*a(n-8). - _Seiichi Manyama_, Aug 10 2024
%e A246883 G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 5*x^5 + 10*x^6 + 17*x^7 +...
%e A246883 where, by definition,
%e A246883 A(x) = 1 + x*(1 + x^3) + x^2*(1 + 2^2*x^3 + x^6)
%e A246883 + x^3*(1 + 3^2*x^3 + 3^2*x^6 + x^9)
%e A246883 + x^4*(1 + 4^2*x^3 + 6^2*x^6 + 4^2*x^9 + x^12)
%e A246883 + x^5*(1 + 5^2*x^3 + 10^2*x^6 + 10^2*x^9 + 5^2*x^12 + x^15) +...
%e A246883 which is also given by the series identity:
%e A246883 A(x) = 1/(1-x+x^4) + 2*x^4/(1-x+x^4)^3 + 6*x^8/(1-x+x^4)^5 + 20*x^12/(1-x+x^4)^7 + 70*x^16/(1-x+x^4)^9 + 252*x^20/(1-x+x^4)^11 + 924*x^24/(1-x+x^4)^13 +...
%e A246883 The logarithm of the g.f. begins:
%e A246883 log(A(x)) = x*(1 + x^3) + x^2*(1 + 6*x^3 + x^6)/2
%e A246883 + x^3*(1 + 15*x^3 + 15*x^6 + x^9)/3
%e A246883 + x^4*(1 + 28*x^3 + 70*x^6 + 28*x^9 + x^12)/4
%e A246883 + x^5*(1 + 45*x^3 + 210*x^6 + 210*x^9 + 45*x^12 + x^15)/5 +...
%e A246883 more explicitly,
%e A246883 log(A(x)) = x + x^2/2 + x^3/3 + 5*x^4/4 + 16*x^5/5 + 31*x^6/6 + 50*x^7/7 + 77*x^8/8 + 145*x^9/9 + 306*x^10/10 + 628*x^11/11 + 1199*x^12/12 +...
%e A246883 where the logarithmic derivative equals
%e A246883 A'(x)/A(x) = (1-x+4*x^3+5*x^4-4*x^7)/((1-x+2*x^2+x^4)*(1-x-2*x^2+x^4)).
%t A246883 CoefficientList[Series[1/Sqrt[(1 - x + x^4)^2 - 4 x^4], {x, 0, 40}], x] (* _Michael De Vlieger_, Sep 10 2021 *)
%o A246883 (PARI) /* By definition: */
%o A246883 {a(n)=local(A=1); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*x^(3*k)) +x*O(x^n)); polcoeff(A, n)}
%o A246883 for(n=0, 40, print1(a(n), ", "))
%o A246883 (PARI) /* From closed formula: */
%o A246883 {a(n)=local(A=1); A= 1/sqrt((1 - x + x^4)^2 - 4*x^4 +x*O(x^n)); polcoeff(A, n)}
%o A246883 for(n=0, 40, print1(a(n), ", "))
%o A246883 (PARI) /* From a series identity: */
%o A246883 {a(n)=local(A=1+x); A=sum(m=0, n, (2*m)!/(m!)^2 * x^(4*m) / (1 - x + x^4 +x*O(x^n))^(2*m+1)); polcoeff(A, n)}
%o A246883 for(n=0, 40, print1(a(n), ", "))
%o A246883 (PARI) /* From a binomial series identity: */
%o A246883 {a(n)=local(A=1+x); A=sum(m=0, n, x^m*(1-x^3)^(2*m+1)*sum(k=0, n, binomial(m+k, k)^2*x^(3*k)) +x*O(x^n)); polcoeff(A, n)}
%o A246883 for(n=0, 40, print1(a(n), ", "))
%o A246883 (PARI) /* From a binomial series identity: */
%o A246883 {a(n)=local(A=1+x); A=sum(m=0, n\4, x^(4*m)*sum(k=0, n-4*m, binomial(m+k, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)}
%o A246883 for(n=0, 40, print1(a(n), ", "))
%o A246883 (PARI) /* From a binomial series identity: */
%o A246883 {a(n)=local(A=1+x); A=sum(m=0, n\4, x^(4*m) * sum(k=0, m, binomial(m, k)^2*x^k) / (1-x +x*O(x^n))^(2*m+1) ); polcoeff(A, n)}
%o A246883 for(n=0, 40, print1(a(n), ", "))
%o A246883 (PARI) /* From exponential formula: */
%o A246883 {a(n)=local(A=1); A=exp(sum(m=1, n, sum(k=0, m, binomial(2*m, 2*k)*x^(3*k)) * x^m/m) +x*O(x^n)); polcoeff(A, n)}
%o A246883 for(n=0, 40, print1(a(n), ", "))
%o A246883 (PARI) /* From formula for a(n): */
%o A246883 {a(n)=sum(k=0, n\3, binomial(n-3*k, k)^2)}
%o A246883 for(n=0, 40, print1(a(n), ", "))
%Y A246883 Cf. A072223, A181665, A246840, A246884, A248193.
%K A246883 nonn
%O A246883 0,5
%A A246883 _Paul D. Hanna_, Sep 06 2014