A182399 - OEIS

%I #33 Aug 22 2021 13:32:15

%S 1,1,1,3,7,21,61,187,583,1837,5885,19027,62167,204917,680621,2275211,

%T 7648519,25852573,87812093,299349795,1023570647,3515918501,

%U 12140103149,41894710427,143835281351,501071173901,1808088546557,6212411239539,17720665594455

%N G.f. A(x) satisfies: A(A(x)) - A(A(x))^2 = x + x^2.

%C a(33) is the first negative term.

%C If B(x) = x + 2*x^2 + 8*x^3 + 36*x^4 + 160*x^5 + 736*x^6 + 3648*x^7 + ..., then g.f. A(x) = x + B(x * A(x)). - _Michael Somos_, Jun 27 2017

%H Paul D. Hanna, <a href="/A182399/b182399.txt">Table of n, a(n) for n = 1..256</a>

%F G.f. satisfies: A(-A(-x)) = x.

%F G.f. satisfies: A(A(x)) = (1 - sqrt(1-4*(x+x^2)))/2 is the g.f. of A025227; thus, A(A(x)) = C(x+x^2) where C(x-x^2) = x.

%F G.f. satisfies: A(-C(-x)) = -I*G(I*x) where C(x-x^2) = x and G(x) is the g.f. of A179270 such that the inverse of function G(x) + I*G(x)^2 equals the complex conjugate: G(x) - I*G(x)^2.

%F a(n) = T(n,1), with T(n, m) = (sum((binomial(k+m,n-k-m)*binomial(2*k+m-1,k+m-1))/(k+m),k,0,n-m)*m -sum(T(n, i) *T(i, m), i, m+1, n-1))/2, n>m, T(n,n) = 1. - _Vladimir Kruchinin_, Apr 28 2012

%e G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 7*x^5 + 21*x^6 + 61*x^7 + 187*x^8 +...

%e Related expansions:

%e A(A(x)) = x + 2*x^2 + 4*x^3 + 12*x^4 + 40*x^5 + 144*x^6 + 544*x^7 + 2128*x^8 +...

%e A(A(x))^2 = x^2 + 4*x^3 + 12*x^4 + 40*x^5 + 144*x^6 + 544*x^7 + 2128*x^8 +...

%e where A(A(x)) - A(A(x))^2 = x + x^2.

%e Let C(x) satisfy C(x-x^2) = x, where C(x) begins:

%e C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 +...+ A000108(n-1)*x^n +...

%e then

%e A(-C(-x)) = x + x^3 + 4*x^5 + 21*x^7 + 122*x^9 + 758*x^11 + 4958*x^13 +...+ (-1)^(n-1)*A179270(2*n-1)*x^(2*n-1) +...

%o (PARI) {a(n)=local(A=x+x^2,G);for(i=1,n,G=subst(A,x,A+x*O(x^n));A=A+(x+x^2-G+G^2)/2);polcoeff(A,n)}

%o for(n=1,30,print1(a(n),", "))

%o (PARI) /* Faster vectorized version: */

%o {MM=100;A=[1];B=x;C=(1-sqrt(1-4*(x+x^2+x*O(x^MM))))/2; for(n=1,oo,A=concat(A,0);B=x*Ser(A); A[n]=Vec((B+subst(C+x*O(x^n),x,serreverse(B)))/2)[n]; print1(A[n],", "))}

%o (PARI) /* PARI/GP Version of Vladimir Kruchinin's formula: */

%o {T(n, m)=if(n==m,1, if(n>m, (sum(k=0,n-m,(binomial(k+m,n-k-m)*binomial(2*k+m-1,k+m-1))/(k+m))*m - sum(i=m+1,n-1,T(n, i) *T(i, m)))/2 ))}

%o {a(n)=T(n,1)}

%o (Maxima) T(n, m):= if n=m then 1 else ((sum((binomial(k+m,n-k-m)*binomial(2*k+m-1,k+m-1))/(k+m),k,0,n-m))*m -sum(T(n, i) *T(i, m), i, m+1, n-1))/2;

%o makelist(T(n, 1), n, 1, 10); /* _Vladimir Kruchinin_, Apr 28 2012 */

%Y Cf. A025227, A179270, A000108.

%K sign

%O 1,4

%A _Paul D. Hanna_, Apr 27 2012