(Translated by https://www.hiragana.jp/)
# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a246653 Showing 1-1 of 1 %I A246653 #9 Sep 02 2014 03:14:27 %S A246653 1,1,3,9,25,87,287,993,3519,12525,45369,165519,608569,2253307,8386575, %T A246653 31370297,117834837,444258387,1680516731,6375706325,24253227159, %U A246653 92481509389,353417696625,1353280137975,5191349266275,19948136148837,76771074483837,295880290515411,1141860205058433,4412129801125275 %N A246653 G.f.: Sum_{n>=0} x^n / (1-x^2)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k] * [Sum_{k=0..n} C(n,k)^2 * x^(2*k)]. %F A246653 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(n-k) * Sum_{j=0..k} C(k,j)^2 * x^j. %F A246653 a(n) = Sum_{k=0..[n/2]} binomial(2*k, k) * binomial(n-k, k)^2. %F A246653 Recurrence: n^2*(4*n-9)*a(n) = (8*n^3 - 26*n^2 + 18*n - 5)*a(n-1) + (28*n^3 - 119*n^2 + 162*n - 75)*a(n-2) + 2*(16*n^3 - 84*n^2 + 142*n - 75)*a(n-3) - 4*(2*n-5)^2*(4*n-5)*a(n-4). - _Vaclav Kotesovec_, Sep 02 2014 %F A246653 a(n) ~ 2^(2*n+1/2) / (Pi*n). - _Vaclav Kotesovec_, Sep 02 2014 %e A246653 G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 25*x^4 + 87*x^5 + 287*x^6 + 993*x^7 +... %e A246653 where the g.f. is given by the binomial series: %e A246653 A(x) = 1/(1-x^2) + x/(1-x^2)^3 * (1 + x) * (1 + x^2) %e A246653 + x^2/(1-x^2)^5 * (1 + 2^2*x + x^2) * (1 + 2^2*x^2 + x^4) %e A246653 + x^3/(1-x^2)^7 * (1 + 3^2*x + 3^2*x^2 + x^3) * (1 + 3^2*x^2 + 3^2*x^4 + x^6) %e A246653 + x^4/(1-x^2)^9 * (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4) * (1 + 4^2*x^2 + 6^2*x^4 + 4^2*x^6 + x^8) %e A246653 + x^5/(1-x^2)^11 * (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5) * (1 + 5^2*x^2 + 10^2*x^4 + 10^2*x^6 + 5^2*x^8 + x^10) +... %t A246653 Table[Sum[Binomial[2*k, k] * Binomial[n-k, k]^2,{k,0,Floor[n/2]}],{n,0,20}] (* _Vaclav Kotesovec_, Sep 02 2014 *) %o A246653 (PARI) /* By definition: */ %o A246653 {a(n)=local(A=1); A=sum(m=0, n, x^m/(1-x^2)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * x^k) * sum(k=0, m, binomial(m, k)^2 * x^(2*k)) +x*O(x^n)); polcoeff(A, n)} %o A246653 for(n=0, 30, print1(a(n), ", ")) %o A246653 (PARI) /* By a binomial identity: */ %o A246653 {a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * x^(m-k) * sum(j=0, k, binomial(k, j)^2 * x^j )+x*O(x^n))), n)} %o A246653 for(n=0, 30, print1(a(n), ", ")) %o A246653 (PARI) /* From a formula for a(n): */ %o A246653 {a(n)=sum(k=0, n\2, binomial(2*k, k)*binomial(n-k, k)^2)} %o A246653 for(n=0, 30, print1(a(n), ", ")) %Y A246653 Cf. A246563. %K A246653 nonn %O A246653 0,3 %A A246653 _Paul D. Hanna_, Aug 31 2014 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE