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%I #8 Oct 16 2024 21:53:12
%S 1,1,4,18,102,684,5216,44388,415672,4234904,46525992,547327904,
%T 6854491840,90940138256,1272862982272,18728235407712,288765445378272,
%U 4653013453323968,78164063007644288,1365903793778043712,24781386644286473856,465969812835308934272,9066115469486822859392
%N G.f.: Sum_{n>=0} x^n / Product_{k=1..2*n} (1 - k*x).
%C Compare to o.g.f. of Bell numbers: Sum_{n>=0} x^n / Product_{k=0..n} (1 - k*x).
%H Vaclav Kotesovec, <a href="/A229286/b229286.txt">Table of n, a(n) for n = 0..160</a>
%H Ira M. Gessel, <a href="https://mathoverflow.net/a/462813/231922">General case of the some R-recursions</a>, answer to question on MathOverflow (2024).
%F 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
%e G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 102*x^4 + 684*x^5 + 5216*x^6 +...
%e where
%e 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)) +...
%o (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)}
%o for(n=0,30,print1(a(n),", "))
%o (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
%Y Cf. A229285.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 18 2013