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A227846 Self-convolution equals A227845.

%I A227846 #19 Sep 29 2019 11:11:17
%S A227846 1,1,3,11,47,215,1029,5077,25615,131455,683749,3595341,19075913,
%T A227846 101978321,548683499,2968535115,16138308655,88107960847,482839660509,
%U A227846 2654879900085,14641704563577,80968432526961,448856443832643,2493858308981331,13884356040349161,77445573778294041
%N A227846 Self-convolution equals A227845.
%H A227846 Vaclav Kotesovec, <a href="/A227846/b227846.txt">Table of n, a(n) for n = 0..430</a>
%F A227846 G.f.: 1 / sqrt( AGM((1+x)^2, (1+x)^2 - 8*x) ), where AGM denotes the arithmetic-geometric mean. - _Paul D. Hanna_, Jul 31 2014
%F A227846 G.f.: sqrt( Sum_{n>=0} A227845(n)*x^n ), where A227845(n) = Sum_{k=0..[n/2]} Sum_{j=k..n-k} binomial(n-k,j)^2*binomial(j,k)^2.
%F A227846 a(n) ~ (1+sqrt(2))^(2*n+1) / (4*n*sqrt(Pi*log(n))) * (1 - (2*gamma + 5*log(2))/(4*log(n)) + (12*gamma^2 + 60*gamma*log(2) + 75*log(2)^2 - 2*Pi^2) / (32*log(n)^2)), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Sep 29 2019
%e A227846 G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 47*x^4 + 215*x^5 + 1029*x^6 +...
%e A227846 where A(x)^2 equals the g.f. of A227845:
%e A227846 A(x)^2 = 1/(1-x) + x/(1-x)^3*(1+x)^2 + x^2/(1-x)^5*(1 + 2^2*x + x^2)^2
%e A227846 + x^3/(1-x)^7*(1 + 3^2*x + 3^2*x^2 + x^3)^2
%e A227846 + x^4/(1-x)^9*(1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^2
%e A227846 + 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 A227846 + 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 +...
%e A227846 and also:
%e A227846 A(x)^2 = 1 + x*(1 + (1+x)) + x^2*(1 + 2^2*(1+x) + (1+2^2*x+x^2)) + x^3*(1 + 3^2*(1+x) + 3^2*(1+2^2*x+x^2) + (1+3^2*x+3^2*x^2+x^3)) + x^4*(1 + 4^2*(1+x) + 6^2*(1+2^2*x+x^2) + 4^2*(1+3^2*x+3^2*x^2+x^3) + (1+4^2*x+6^2*x^2+4^2*x^3+x^4)) + ...
%e A227846 Explicitly,
%e A227846 A(x)^2 = 1 + 2*x + 7*x^2 + 28*x^3 + 125*x^4 + 590*x^5 + 2891*x^6 +...+ A227845(n)*x^n +...
%o A227846 (PARI) {a(n)=polcoeff(sqrt(sum(m=0,n,x^m*sum(k=0,m,binomial(m,k)^2*sum(j=0,k,binomial(k,j)^2*x^j)+x*O(x^n)))),n)}
%o A227846 for(n=0,30,print1(a(n),", "))
%o A227846 (PARI) /* From g.f. 1/sqrt( AGM((1+x)^2, (1+x)^2 - 8*x) ): */
%o A227846 {a(n)=local(A); A = 1/sqrt( agm((1+x)^2, (1+x)^2 - 8*x +x*O(x^n))); polcoeff(A, n)}
%o A227846 for(n=0, 30, print1(a(n), ", "))
%Y A227846 Cf. A227845.
%K A227846 nonn
%O A227846 0,3
%A A227846 _Paul D. Hanna_, Aug 01 2013