A097090 G.f. A(x) satisfies: A(A(x)) = x*(1+2*x)^2.

1, 2, -2, 6, -22, 80, -228, 18, 6694, -65604, 396804, -1336332, -2510244, 64799884, -302351144, -1410221598, 23754923526, 16833211660, -2598949277964, 14767224078420, 229529725999500, -3478598282993328, -13287810766972728, 667271251276705140, -1630867775606147844

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..130

Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$, arXiv:1302.1986

Formula

a(n)=T(n,1), T(n,m)=if n=m then 1 else 1/2*(binomial(2*m,n-m)*2^(n-m)-sum(i=m+1..n-1, T(n,i)*T(i,m))). [From Vladimir Kruchinin, Nov 10 2011]

Example

G.f.: A(x) = x + 2*x^2 - 2*x^3 + 6*x^4 - 22*x^5 + 80*x^6 - 228*x^7 + 18*x^8 +...
where A(A(x)) = x + 4*x^2 + 4*x^3.
Illustrate Vladimir Kruchinin's formula by the triangular matrix T:
1;
2, 1;
-2, 4, 1;
6, 0, 6, 1;
-22, 4, 6, 8, 1;
80, -16, 2, 16, 10, 1;
-228, 48, -6, 8, 30, 12, 1;
18, -12, 0, 0, 30, 48, 14, 1;
6694, -1460, 232, -32, 10, 76, 70, 16, 1; ...
in which the g.f. of column k = A(x)^k and A(x) is the g.f. of this sequence.
The matrix square, T^2, of the above triangle begins:
1;
4, 1;
4, 8, 1;
0, 24, 12, 1;
0, 32, 60, 16, 1;
0, 16, 160, 112, 20, 1;
0, 0, 240, 448, 180, 24, 1;
0, 0, 192, 1120, 960, 264, 28, 1;
0, 0, 64, 1792, 3360, 1760, 364, 32, 1; ...
in which the g.f. column k = (x + 4*x^2 + 4*x^3)^k = sum(n>=k, binomial(2*k,n-k)*2^(n-k)*x^n).

Mathematica

T[n_, m_] := T[n, m] = If[n == m, 1, (1/2)*(Binomial[2 m, n - m]*2^(n - m) - Sum[T[n, i]*T[i, m], {i, m+1, n-1}])]; a[n_] := T[n, 1] // Numerator; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Oct 29 2015, after Vladimir Kruchinin *)

Prog

(PARI) {a(n)=local(A, B, F); F=x*(1+2*x+x*O(x^n))^2; A=F; for(i=0, n, B=serreverse(A); A=(A+subst(B, x, F))/2); polcoeff(A, n, x)}
(PARI) /* Vladimir Kruchinin's formula: a(n) = T(n, 1) where: */
{T(n, k)=if(n==k, 1, 1/2*(binomial(2*k, n-k)*2^(n-k)-sum(j=k+1, n-1, T(n, j)*T(j, k))))}
{a(n)=T(n, 1)} /* Paul D. Hanna, Nov 10 2011 */

Crossrefs

Cf. A107099.

Sequence in context: A271217 A202743 A007985 * A085403 A112478 A184715

Adjacent sequences: A097087 A097088 A097089 * A097091 A097092 A097093

Keyword

sign

Author

Paul D. Hanna, Jul 23 2004

Status

approved