This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A322244 #22 Aug 18 2025 09:58:50
%S A322244 1,3,41,315,3345,31923,328889,3337323,34600225,359225955,3760299081,
%T A322244 39497556123,416692693041,4409256847635,46791791441625,
%U A322244 497734241873355,5305782027097665,56663444325365955,606142658305541225,6493612892317230075,69658589316520324945,748141936546712050035,8043908203413946807545,86573015247061060850475,932597459464760512144225
%N A322244 G.f.: 1/sqrt(1 - 6*x - 55*x^2).
%H A322244 Seiichi Manyama, <a href="/A322244/b322244.txt">Table of n, a(n) for n = 0..962</a>
%F A322244 a(n) = Sum_{k=0..n} 11^(n-k) * (-4)^k * binomial(n,k)*binomial(2*k,k).
%F A322244 a(n) = Sum_{k=0..n} (-5)^(n-k) * 4^k * binomial(n,k)*binomial(2*k,k).
%F A322244 a(n) equals the (central) coefficient of x^n in (1 + 3*x + 16*x^2)^n.
%F A322244 a(n) ~ 11^(n + 1/2) / (4*sqrt(Pi*n)). - _Vaclav Kotesovec_, Dec 13 2018
%F A322244 D-finite with recurrence: n*a(n) +3*(-2*n+1)*a(n-1) +55*(-n+1)*a(n-2)=0. - _R. J. Mathar_, Jan 16 2020
%F A322244 a(n) = (1/4)^n * Sum_{k=0..n} (-5)^k * 11^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - _Seiichi Manyama_, Aug 18 2025
%e A322244 G.f.: A(x) = 1 + 3*x + 41*x^2 + 315*x^3 + 3345*x^4 + 31923*x^5 + 328889*x^6 + 3337323*x^7 + 34600225*x^8 + 359225955*x^9 + 3760299081*x^10 + ...
%e A322244 such that A(x)^2 = 1/(1 - 6*x - 55*x^2).
%e A322244 RELATED SERIES.
%e A322244 exp( Sum_{n>=1} a(n)*x^n/n ) = 1 + 3*x + 25*x^2 + 171*x^3 + 1457*x^4 + 12243*x^5 + 109769*x^6 + 997755*x^7 + 9314657*x^8 + 88177059*x^9 + 847159161*x^10 + ...
%t A322244 a[n_] := Sum[(-5)^(n-k) * 4^k * Binomial[n,k] * Binomial[2k,k], {k, 0, n}]; Array[a, 20, 0] (* _Amiram Eldar_, Dec 13 2018 *)
%t A322244 CoefficientList[Series[1/Sqrt[1-6x-55x^2],{x,0,40}],x] (* _Harvey P. Dale_, Aug 13 2024 *)
%o A322244 (PARI) /* Using generating function: */
%o A322244 {a(n) = polcoeff( 1/sqrt(1 - 6*x - 55*x^2 +x*O(x^n)),n)}
%o A322244 for(n=0,30,print1(a(n),", "))
%o A322244 (PARI) /* Using binomial formula: */
%o A322244 {a(n) = sum(k=0,n, (-5)^(n-k)*4^k*binomial(n,k)*binomial(2*k,k))}
%o A322244 for(n=0,30,print1(a(n),", "))
%o A322244 (PARI) /* Using binomial formula: */
%o A322244 {a(n) = sum(k=0,n, 11^(n-k)*(-4)^k*binomial(n,k)*binomial(2*k,k))}
%o A322244 for(n=0,30,print1(a(n),", "))
%o A322244 (PARI) /* a(n) is central coefficient in (1 + 3*x + 4*x^2)^n */
%o A322244 {a(n) = polcoeff( (1 + 3*x + 16*x^2 +x*O(x^n))^n, n)}
%o A322244 for(n=0,30,print1(a(n),", "))
%Y A322244 Cf. A322245 (a(n)^2).
%K A322244 nonn,easy
%O A322244 0,2
%A A322244 _Paul D. Hanna_, Dec 09 2018