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A316705 E.g.f.: Sum_{n>=0} x^n/n! * Product_{k=1..n} (2*n+1-2*k) + (2*k-1)*x.

%I A316705 #15 Jul 18 2018 06:57:46
%S A316705 1,1,5,45,513,7745,142485,3095421,77642145,2207145825,70130493765,
%T A316705 2463100122285,94752421655265,3962161404127329,178943401595685909,
%U A316705 8680576995359894205,450150904632193002945,24850264116962803786305,1455015398837011003805445,90062955077484708745769133,5876178416626462668682616385,403059737428052979318873127425
%N A316705 E.g.f.: Sum_{n>=0} x^n/n! * Product_{k=1..n} (2*n+1-2*k) + (2*k-1)*x.
%H A316705 Paul D. Hanna, <a href="/A316705/b316705.txt">Table of n, a(n) for n = 0..300</a>
%F A316705 E.g.f.: A(x) = sqrt( (1 + 2*x*W(x))*(1 + 2*x^2*W(x)) ) such that W(x) satisfies: W(x)/(1 + 2*x*W(x)) = ( W(x)/(1 + 2*x^2*W(x)) )^x.
%F A316705 a(n)/n! ~ c * d^n / sqrt(n), where d = 3.346513389529679772056152566067040813392... and c = 0.34882587166136471331152567... - _Vaclav Kotesovec_, Jul 18 2018
%e A316705 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 45*x^3/3! + 513*x^4/4! + 7745*x^5/5! + 142485*x^6/6! + 3095421*x^7/7! + 77642145*x^8/8! + 2207145825*x^9/9! + 70130493765*x^10/10! + ...
%e A316705 such that
%e A316705 A(x) = 1 + (1 + x)*x + (3 + x)*(1 + 3*x)*x^2/2! + (5 + x)*(3 + 3*x)*(1 + 5*x)*x^3/3! + (7 + x)*(5 + 3*x)*(3 + 5*x)*(1 + 7*x)*x^4/4! + (9 + x)*(7 + 3*x)*(5 + 5*x)*(3 + 7*x)*(1 + 9*x)*x^5/5! + ...
%e A316705 Also,
%e A316705 A(x) = sqrt( (1 + 2*x*W(x))*(1 + 2*x^2*W(x)) )
%e A316705 where
%e A316705 W(x) = 1 + (1 + x)*(2*x) + (2 + x)*(1 + 2*x)*(2*x)^2/2! + (3 + x)*(2 + 2*x)*(1 + 3*x)*(2*x)^3/3! + (4 + x)*(3 + 2*x)*(2 + 3*x)*(1 + 4*x)*(2*x)^4/4! + (5 + x)*(4 + 2*x)*(3 + 3*x)*(2 + 4*x)*(1 + 5*x)*(2*x)^5/5! + ...
%e A316705 Explicitly,
%e A316705 W(x) = 1 + 2*x + 12*x^2/2! + 108*x^3/3! + 1312*x^4/4! + 20320*x^5/5! + 381408*x^6/6! + 8420160*x^7/7! + 213813248*x^8/8! + ... + A316704(n)*x^n/n! + ...
%e A316705 where W(x) satisfies
%e A316705 W(x)/(1 + 2*x*W(x)) = ( W(x)/(1 + 2*x^2*W(x)) )^x.
%o A316705 (PARI) {a(n) = my(A); A = sum(m=0,n, x^m/m! * prod(k=1,m, 2*m+1-2*k + (2*k-1)*x +x*O(x^n))); n!*polcoeff(A,n)}
%o A316705 for(n=0,30, print1(a(n),", "))
%Y A316705 Cf. A316704, A316370.
%K A316705 nonn
%O A316705 0,3
%A A316705 _Paul D. Hanna_, Jul 16 2018