A222013 - OEIS

%I #8 Dec 06 2024 08:47:26

%S 1,1,2,7,32,172,1038,6865,49098,376942,3094812,27129690,253821716,

%T 2534600760,27012498668,307083883519,3719224056464,47898505899624,

%U 654343988611350,9455986402701388,144138413744793426,2311030293590097634,38871924229882607774

%N 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).

%C Compare the g.f. to the identities:

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

%C (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).

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

%H Paul D. Hanna, <a href="/A222013/b222013.txt">Table of n, a(n) for n = 0..160</a>

%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 32*x^4 + 172*x^5 + 1038*x^6 +...

%e where

%e 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)) +...

%o (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)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A222014, A221586.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 04 2013