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URL: https://oeis.org/A229286

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A229286
Expansion of Sum_{n>=0} x^n / Product_{k=1..2*n} (1 - k*x).
7
1, 1, 4, 18, 102, 684, 5216, 44388, 415672, 4234904, 46525992, 547327904, 6854491840, 90940138256, 1272862982272, 18728235407712, 288765445378272, 4653013453323968, 78164063007644288, 1365903793778043712, 24781386644286473856, 465969812835308934272, 9066115469486822859392
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
Ira M. Gessel, General case of the some R-recursions, answer to question on MathOverflow (2024).
FORMULA
a(n) = Sum_{k=0..n} Stirling2(n+k,2*k). - Seiichi Manyama, Jan 21 2026
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), ", "))
CROSSREFS
Sequence in context: A327833 A350267 A064852 * A191365 A335459 A159666
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 18 2013
STATUS
approved