OFFSET
0,3
COMMENTS
Compare to o.g.f. of Bell numbers: Sum_{n>=0} x^n / Product_{k=0..n} (1 - k*x).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..160
Ira M. Gessel, General case of the some R-recursions, answer to question on MathOverflow (2024).
FORMULA
a(n) = R(n-1, 0) for n > 0 with a(0) = 1 where R(n, q) = (4*q+3)*R(n-1, q) - 2*(q+1)*(2*q+1)*R(n-2, q) + R(n-1, q+1) for n > 0, q >= 0 with R(n, q) = [n = 0] for n < 1, q >= 0. - Mikhail Kurkov, Oct 02 2024
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 102*x^4 + 684*x^5 + 5216*x^6 +...
where
A(x) = 1 + x/((1-x)*(1-2*x)) + x^2/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)) + x^3/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)*(1-6*x)) +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=0, n, x^m/prod(k=1, 2*m, 1-k*x+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) upto(n) = my(v1, v2, v3, v4); v1 = vector(n, i, 1); v2 = vector(n, i, 0); v3 = vector(n+1, i, 0); for(i=1, min(n+1, 2), v3[i] = 1); for(i=1, n-1, v4 = v1; for(j=1, n-i, v1[j] = (4*j-1)*v1[j] - 2*j*(2*j-1)*v2[j] + v1[j+1]); v3[i+2] = v1[1]; v2 = v4); v3 \\ Mikhail Kurkov, Oct 02 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 18 2013
STATUS
approved