A128709 - OEIS

%I #22 Feb 08 2022 20:19:09

%S 1,1,4,31,364,5746,113944,2719291,75843724,2420160286,86941080904,

%T 3471911602006,152562875644984,7315129181611876,380045172886143664,

%U 21266347877729314771,1275148311699896290444,81563275661324271278566

%N O.g.f.: A(x) = 1/(1-1*x/(1-3*x/(1-5*x/(1-7*x/(1-...-(2n-1)*x/(1-...)))))) (continued fraction).

%C Hankel transform is A168440. - _Paul Barry_, Nov 25 2009

%F a(n) = Sum_{k=0..n} (-1)^k*2^(n-k)*A053979(n,k). - _Philippe Deléham_, Mar 24 2007

%F a(n) = Sum_{k=0..n} A094344(n,k)*3^(n-k). - _Philippe Deléham_, Mar 27 2007

%F G.f.: 1/(1-x-3x^2/(1-8x-35x^2/(1-16x-99x^2/(1-24x-195x^2/(1-32x-323x^2/(1-... (continued fraction). - _Paul Barry_, Nov 25 2009

%F a(n) = top left term of M^n, n > 0; M = the infinite square production matrix:

%F   1, 3, 0, 0, ...

%F   1, 3, 5, 0, ...

%F   1, 3, 5, 7, ...

%F   ...

%F Also, a(n+1) = sum of top row terms of M^n. Example: top row of M^3 = (31, 93, 135, 105, 0, 0, 0, ...), where a(3) = 31 and a(4) = 364 = (31 + 93 + 135 + 105). - Gary W. Adamson, Jul 14 2011

%F G.f.: 1/T(0) where T(k) =  1 - x*(4*k+1)/(1 - x*(4*k+3)/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 19 2013

%F G.f.: G(0), where G(k) = 1 - x*(2*k+1)/(x*(2*k+1) - 1/G(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Aug 05 2013

%F a(n) ~ 2^(2*n - 1/2) * (n-1)! / Pi. - _Vaclav Kotesovec_, Aug 24 2017

%e G.f.: A(x) = 1 + x + 4x^2 + 31x^3 + 364x^4 + 5746x^5 + ...;

%e A(x) = 1/(1 - x*(1 + 3x + 24x^2 + 297x^3 + 4896x^4 + ...));

%e A(x) = 1/(1 - x/(1 - 3x*(1 + 5x + 60x^2 + 1035x^3 + 22500x^4 + ...)));

%e A(x) = 1/(1 - x/(1 - 3x/(1 - 5x*(1 + 7x + 112x^2 + 2485x^3 + ...)))).

%t nmax = 20; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[(2*Range[nmax + 1]-1)*x]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 24 2017 *)

%o (PARI) {a(n)=local(CF=1+x*O(x^n));for(k=0,n,CF=1/(1-(2*n-2*k+1)*x*CF));polcoeff(CF,n,x)}

%K nonn

%O 0,3

%A _Paul D. Hanna_, Mar 23 2007