A182399 G.f. A(x) satisfies: A(A(x)) - A(A(x))^2 = x + x^2.

1, 1, 1, 3, 7, 21, 61, 187, 583, 1837, 5885, 19027, 62167, 204917, 680621, 2275211, 7648519, 25852573, 87812093, 299349795, 1023570647, 3515918501, 12140103149, 41894710427, 143835281351, 501071173901, 1808088546557, 6212411239539, 17720665594455

Offset

1,4

Comments

a(33) is the first negative term.

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

Links

Paul D. Hanna, Table of n, a(n) for n = 1..256

Formula

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

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.

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.

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

Example

G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 7*x^5 + 21*x^6 + 61*x^7 + 187*x^8 +...
Related expansions:
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 +...
A(A(x))^2 = x^2 + 4*x^3 + 12*x^4 + 40*x^5 + 144*x^6 + 544*x^7 + 2128*x^8 +...
where A(A(x)) - A(A(x))^2 = x + x^2.
Let C(x) satisfy C(x-x^2) = x, where C(x) begins:
C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 +...+ A000108(n-1)*x^n +...
then
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) +...

Prog

(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)}
for(n=1, 30, print1(a(n), ", "))
(PARI) /* Faster vectorized version: */
{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], ", "))}
(PARI) /* PARI/GP Version of Vladimir Kruchinin's formula: */
{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 ))}
{a(n)=T(n, 1)}
(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;
makelist(T(n, 1), n, 1, 10); /* Vladimir Kruchinin, Apr 28 2012 */

Crossrefs

Cf. A025227, A179270, A000108.

Sequence in context: A122983 A351822 A005355 * A025235 A129366 A270049

Adjacent sequences: A182396 A182397 A182398 * A182400 A182401 A182402

Keyword

sign

Author

Paul D. Hanna, Apr 27 2012

Status

approved