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, 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

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

Paul D. Hanna, Table of n, a(n) for n = 0..400

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

Cf. A000108.

Sequence in context: A073680 A079557 A090709 * A112279 A130166 A007097

Adjacent sequences: A357217 A357218 A357219 * A357221 A357222 A357223

Keyword

nonn

Author

Paul D. Hanna, Oct 16 2022

Status

approved