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Revision History for A004212
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FORMULA
Touchard's congruence holds: for prime p not equal to 3, a(p+k) = = (a(k) + a(k+1)) (mod p) for k = 0,1,2,... (adapt the proof of Theorem 10.1 in Gessel). In particular, a(p) = = 2 (mod p) for prime p <> 3. (End)
FORMULA
E.g.f.: exp(int(Integral_{t = 0..x, } exp(3*t))). - Joerg Arndt, Apr 30 2011
Recurrence equation: a(n+1) = Sum_{k = 0..n} 3^(n-k)*C(n,k)*a(k). Written umbrally this is a(n+1) = (a + 3)^n (expand the binomial and replace a^k with a(k)). More generally, a*f(a) = f(a+3) holds umbrally for any polynomial f(x). An inductive argument then establishes the umbral recurrence a*(a-3)*(a-6)*...*(a-3*(n-1)) = 1 with a(0) = 1. Compare with the Bell numbers B(n) = A000110(n), which satisfy the umbral recurrence B*(B-1)*...*(B-(n-1)) = 1 with B(0) = 1.
Touchard's congruence holds : for prime p not equal to 3: , a(p+k) = (a(k) + a(k+1)) (mod p) for k = 0,1,2,... (adapt the proof of Theorem 10.1 in Gessel). In particular, a(p) = 2 (mod p) for prime p <> 3. (End)
(End)
a(n) = exp(-1/3)*Sum_{n >= 0} (3*n)^k/(n!*3^n). - Peter Bala, Jun 29 2024
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FORMULA
a(n) ~ 3^n * n^n * exp(n/LambertW(3*n) - 1/3 - n) / (sqrt(1 + LambertW(3*n)) * LambertW(3*n)^n). - Vaclav Kotesovec, Jul 15 2021
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FORMULA
G.f. satisfies A(x) = 1+x/(1-3*x)*A(3*x/(1-3*x)). a(n) = Sum_{k=1..n} 3^(n-k)*binomial(n-1,k-1)*a(k-1), n > 0, a(0)=1. - Vladimir Kruchinin, Nov 28 2011 [corrected by _Ilya Gutkovskiy_, May 02 2019]
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