|
| |
|
|
A120588
|
|
G.f. satisfies: 3*A(x) = 2 + x + A(x)^2, starting with [1,1,1].
|
|
20
|
|
|
|
1, 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, 18367353072152
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
COMMENTS
|
This is essentially a duplicate of entry A000108, the Catalan numbers (a(n) = A000108(n-1) for n>0).
In order for the g.f. of an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n, where n > 1, it is necessary that the sequence start with [1, d, m*n*(n-1)/2], where d divides m*n*(n-1)/2 (m>0) and that the coefficients are given by r = n + d^2/m, c = r-1 and b = d^3/m. The remaining terms may then be integer and still satisfy: a_n(k) = r*a(k), where a_n(k) is the k-th term of the n-th self-convolution of the sequence.
|
|
|
LINKS
|
Table of n, a(n) for n=0..27.
|
|
|
FORMULA
|
G.f.: A(x) = 1 + Series_Reversion(1+3*x - (1+x)^2).
Lagrange Inversion yields g.f.: A(x) = Sum_{n>=0} C(2*n,n)/(n+1)*(2+x)^(n+1)/3^(2*n+1).
G.f.:(3-sqrt(1-4*x))/2 [From Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009]
a(n) = Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
|
|
|
EXAMPLE
|
A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 +...
A(x)^3 = 1 + 2*x + 3*x^2 + 6*x^3 + 15*x^4 + 42*x^5 + 126*x^6 + 396*x^7 +..
More generally, given the functional equation:
r*A(x) = r-1 + b*x + A(x)^n
the series solution is:
A(x) = Sum_{i>=0} C(n*i,i)/(n*i-i+1)*(r-1+bx)^(n*i-i+1)/r^(n*i+1)
which can be expressed as:
A(x) = G( (r-1+bx)^(n-1)/r^n ) * (r-1+bx)/r
where G(x) satisfies: G(x) = 1 + x*G(x)^n .
Also we have:
A(x) = 1 + Series_Reversion[ (1 + r*x - (1+x)^n )/b ].
|
|
|
PROG
|
(PARI) {a(n)=local(A=1+x+x^2+x*O(x^n)); for(i=0, n, A=A-3*A+2+x+A^2); polcoeff(A, n)}
(PARI) {a(n) = local(A); if( n<1, n==0, A = vector(n); A[1] = 1; for( k=2, n,
A[k] = sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */
|
|
|
CROSSREFS
|
Cf. A120589 (A(x)^2); A120590 - A120607.
Sequence in context: A033191 A211216 A115140 * A168491 A000108 A057413
Adjacent sequences: A120585 A120586 A120587 * A120589 A120590 A120591
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Paul D. Hanna, Jun 16 2006, Jan 24 2008
|
|
|
STATUS
|
approved
|
| |
|
|
