|
| |
|
|
A357220
|
|
a(n) = coefficient of x^n in Sum_{n>=0} x^n/(1 - x*C(x)^n), where C(x) = 1/(1 - x*C(x)) is a g.f. of the Catalan numbers (A000108).
|
|
1
|
|
|
|
1, 2, 3, 5, 11, 31, 101, 355, 1304, 4938, 19155, 75857, 306075, 1256782, 5248018, 22278742, 96141427, 421787510, 1881594580, 8537257714, 39408291543, 185114771571, 885043068109, 4307374572585, 21340519926034, 107627435856554, 552473684683454, 2885909702592788
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
Related Identity due to George E. Andrews: Sum_{n>=0} x^(k*n)/(1 - x^k*q^n) = Sum_{n>=0} q^(n^2) * x^(2*k*n) * (1 + x^k*q^n)/(1 - x^k*q^n), which holds for positive integer k.
|
|
|
LINKS
|
|
|
|
FORMULA
|
Generating function A(x) = Sum_{n>=0} a(n)*x^n may be expressed by the following formulas, where C(x) = 1/(1 - x*C(x)).
(1) A(x) = Sum_{n>=0} x^n/(1 - x*C(x)^n).
(2) A(x) = Sum_{n>=0} C(x)^(n^2) * x^(2*n) * (1 + x*C(x)^n)/(1 - x*C(x)^n).
(3) A(x-x^2) = Sum_{n>=0} x^n * (1-x)^(2*n) / ((1-x)^n - x*(1-x)).
|
|
|
EXAMPLE
|
G.f.: A(x) = 1 + 2*x + 3*x^2 + 5*x^3 + 11*x^4 + 31*x^5 + 101*x^6 + 355*x^7 + 1304*x^8 + 4938*x^9 + 19155*x^10 + 75857*x^11 + 306075*x^12 + ...
such that
A(x) = 1/(1 - x) + x/(1 - x*C(x)) + x^2/(1 - x*C(x)^2) + x^3/(1 - x*C(x)^3) + x^4/(1 - x*C(x)^4) + x^5/(1 - x*C(x)^5) + ... + x^n/(1 - x*C(x)^n) + ...
also
A(x) = (1+x)/(1-x) + C(x)*x^2*(1+x*C(x))/(1-x*C(x)) + C(x)^4*x^4*(1+x*C(x)^2)/(1-x*C(x)^2) + C(x)^9*x^6*(1+x*C(x)^3)/(1-x*C(x)^3) + C(x)^16*x^8*(1+x*C(x)^4)/(1-x*C(x)^4) + ... + C(x)^(n^2)*x^(2*n)*(1+x*C(x)^n)/(1-x*C(x)^n) + ...
where C(x) = 1/(1 - x*C(x)) begins
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + 4862*x^9 + 16796*x^10 + A000108(n)*x^n + ...
|
|
|
PROG
|
(PARI) {a(n) = my(A, C = 1/x*serreverse(x - x^2 + O(x^(n+2))));
A = sum(m=0, n+1, x^m/(1 - x*C^m)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
