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A246539 G.f.: Sum_{n>=0} 3^n * x^n / (1-x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k]^2. 4

%I #12 Nov 05 2014 16:58:48

%S 1,4,25,184,1489,12796,114241,1047568,9796057,92989876,893250193,

%T 8663461000,84697699297,833616713164,8251811812465,82088310375904,

%U 820140832103881,8225191769615620,82768982623011841,835404195075128536,8454743911307336857,85775961307556225596,872156269935215409577

%N G.f.: Sum_{n>=0} 3^n * x^n / (1-x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k]^2.

%C a(n) == 1 (mod 3) for n>=0.

%F G.f.: Sum_{n>=0} x^n / (1-3*x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * 3^k * x^k]^2.

%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * 3^k * Sum_{j=0..k} C(k,j)^2 * x^j.

%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * 3^(n-k) * Sum_{j=0..k} C(k,j)^2 * 3^j * x^j.

%F a(n) = Sum_{k=0..[n/2]} 3^k * Sum_{j=0..n-2*k} C(n-k, k+j)^2 * C(k+j, j)^2 * 3^j.

%F Recurrence: (n-3)*n^2*a(n) = 4*(n-3)*(3*n^2 - 3*n + 1)*a(n-1) - 3*(n-1)*(3*n^2 - 12*n + 8)*a(n-2) - 8*(n-2)*(7*n^2 - 28*n + 24)*a(n-3) - 9*(n-3)*(3*n^2 - 12*n + 8)*a(n-4) + 36*(n-1)*(3*n^2 - 21*n + 37)*a(n-5) - 27*(n-4)^2*(n-1)*a(n-6). - _Vaclav Kotesovec_, Nov 05 2014

%F a(n) ~ sqrt(3) * (2 + 2*sqrt(3) + sqrt(13+8*sqrt(3)))^(n+1) / (12*Pi*n). - _Vaclav Kotesovec_, Nov 05 2014

%e G.f.: A(x) = 1 + 4*x + 25*x^2 + 184*x^3 + 1489*x^4 + 12796*x^5 +...

%e where

%e A(x) = 1/(1-x) + 3*x/(1-x)^3*(1+x)^2

%e + 3^2*x^2/(1-x)^5*(1 + 2^2*x + x^2)^2

%e + 3^3*x^3/(1-x)^7*(1 + 3^2*x + 3^2*x^2 + x^3)^2

%e + 3^4*x^4/(1-x)^9*(1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^2

%e + 3^5*x^5/(1-x)^11*(1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)^2

%e + 3^6*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 +...

%t Table[Sum[3^k * Sum[Binomial[n-k, k+j]^2 * Binomial[k+j, j]^2 * 3^j,{j,0,n-2*k}],{k,0,Floor[n/2]}],{n,0,20}] (* _Vaclav Kotesovec_, Nov 05 2014 *)

%o (PARI) /* By definition: */

%o {a(n)=local(A=1); A=sum(m=0, n, 3^m*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)}

%o for(n=0, 25, print1(a(n), ", "))

%o (PARI) /* By a binomial identity: */

%o {a(n)=local(A=1); A=sum(m=0, n, x^m/(1-3*x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * 3^k * x^k)^2 +x*O(x^n)); polcoeff(A, n)}

%o for(n=0, 25, print1(a(n), ", "))

%o (PARI) /* By a binomial identity: */

%o {a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * 3^k * sum(j=0, k, binomial(k, j)^2 * x^j)+x*O(x^n))), n)}

%o for(n=0, 25, print1(a(n), ", "))

%o (PARI) /* By a binomial identity: */

%o {a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * 3^(m-k) * sum(j=0, k, binomial(k, j)^2 * 3^j * x^j)+x*O(x^n))), n)}

%o for(n=0, 25, print1(a(n), ", "))

%o (PARI) /* By a formula for a(n): */

%o {a(n)=sum(k=0, n\2, sum(j=0, n-2*k, 3^k * binomial(n-k, k+j)^2 * binomial(k+j, j)^2 * 3^j))}

%o for(n=0, 25, print1(a(n), ", "))

%Y Cf. A246538, A246540, A246056, A246423, A243948, A245929, A227845, A245925.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 28 2014

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Last modified March 29 05:48 EDT 2024. Contains 371265 sequences. (Running on oeis4.)