A222013 G.f. satisfies: A(x) = Sum_{n>=0} n! * x^n * A(x)^(n*(n+1)/2) / Product_{k=1..n} (1 + k*x*A(x)^k).

1, 1, 2, 7, 32, 172, 1038, 6865, 49098, 376942, 3094812, 27129690, 253821716, 2534600760, 27012498668, 307083883519, 3719224056464, 47898505899624, 654343988611350, 9455986402701388, 144138413744793426, 2311030293590097634, 38871924229882607774

Offset

0,3

Comments

Compare the g.f. to the identities:

(1) 1/(1-x) = Sum_{n>=0} n!*x^n / Product_{k=1..n} (1 + k*x).

(2) C(x) = Sum_{n>=0} n!*x^n*C(x)^n / Product_{k=1..n} (1 + k*x*C(x)), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Conjecture: a(n) is odd iff n = 2^k - 1 for some k >= 0. - Paul D. Hanna, Dec 06 2024

Links

Paul D. Hanna, Table of n, a(n) for n = 0..160

Example

G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 32*x^4 + 172*x^5 + 1038*x^6 +...
where
A(x) = 1 + x*A(x)/(1+x*A(x)) + 2!*x^2*A(x)^3/((1+x*A(x))*(1+2*x*A(x)^2)) + 3!*x^3*A(x)^6/((1+x*A(x))*(1+2*x*A(x)^2)*(1+3*x*A(x)^3)) + 4!*x^4*A(x)^10/((1+x*A(x))*(1+2*x*A(x)^2)*(1+3*x*A(x)^3)*(1+4*x*A(x)^4)) +...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, m!*x^m*A^(m*(m+1)/2)/prod(k=1, m, 1+k*x*(A+x*O(x^n))^k))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A222014, A221586.

Sequence in context: A107593 A125277 A321688 * A179488 A191809 A161392

Adjacent sequences: A222010 A222011 A222012 * A222014 A222015 A222016

Keyword

nonn

Author

Paul D. Hanna, Feb 04 2013

Status

approved