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Munagi, Mar 2. 0 2. Divide and Conquer General method Divide Split the input with nsample points into ksubsets, 1 <k n, with no overlap between subsets Conquer Solve each of the.Triangle (addition) scheme to produce terms, derived from the recurrence, from Oscar Arevalo (loarevalo(AT)sbcglobal. May 1. 1 2. 00. 5: 1 1 2 2 3 5 5 7 1. Then a(n) is the sum of all components. E. g., a(3)=5 because there are exactly five functions that satisfy the two conditions: f. Walsh, Feb 2. 0 2. Number of asynchronic siteswap patterns of length n which have no zero- throws (i. Allen Knutson) is equal to the number of balls. E. g., for n=4, the condition is satisfied by the following 1. Also number of ways to choose n permutations from identity and cyclic permutations (1 2), (1 2 3), .., (1 2 3 .. For n=3 we get the following five: id o id o id, id o (1 2) o (1 2), (1 2) o id o (1 2), (1 2) o (1 2) o id, (1 2 3) o (1 2 3) o (1 2 3). Example: a(3) = 5 counts all permutations on . Example: a(3)=5 counts . If n=1 then the result will be e, about 2. If n=2, the result will be 2e. If n=3, the result will be 5e. This continues, following the pattern of the Bell numbers: e, 2e, 5e, 1. Love (japanada. 11(AT)yahoo. Feb 2. 2 2. 00. 7From Gottfried Helms, Mar 3. Start)This sequence is also the first column in the matrix- exponential of the (lower triangular) Pascal- matrix, scaled by exp(- 1): PE = exp(P) / exp(1) = 1 1 1 2 2 1 5 6 3 1 1. First 4 columns are A0. A0. 33. 30. 6, A1. A1. 05. 48. 0. The general case is mentioned in the two latter entries. PE is also the Hadamard- product Toeplitz(A0. X) P: 1 1 1 2 1 1 5 2 1 1 1. X) P 5. 2 1. End)The terms can also be computed with finite steps and precise integer arithmetic. Instead of exp(P)/exp(1) one can compute A = exp(P - I) where I is the identity- matrix of appropriate dimension since (P- I) is nilpotent to the order of its dimension. Lunnon recently found the Bell pseudoprimes 2. C4. 6 = 3*2. 3*1. Bell pseudoprimes are extremely scarce. So the second Bell pseudoprime is unlikely to be known with certainty in the near future. I confirmed that 2. Wilson, Aug 0. 4 2. Sep 2. 4 2. 00. 7This sequence and A0. A1. 33. 31. 4. Also row sums of triangle A1. Adamson, Jan 2. 1 2. This is the Expoential transform of A0. An n- pattern sequence is a sequence of integers (a. For example, Bell(3) = 5 since the 3- pattern sequences are (1,1,1), (1,1,3), (1,2,1), (1,2,2) and (1,2,3). Bell(n) is the number of sequences of positive integers (N. It is interesting to note that if we strengthen the latter condition to N. Adamson, Dec 0. 4 2. The entries f(i, j) in the exponential of the infinite lower- triangular matrix of binomial coefficients b(i, j) are f(i, j) = b(i, j) e a(i - j). Adamson, Jan 0. 2 2. Equals A1. 54. 10. A0. 14. 18. 2, where A0. A0. 07. 31. 8^(- 1). Starting with offset 1 = A1. A1. 54. 10. 9. Adamson, Jan 0. Repeated iterates of (binomial transform of . Adamson, Jan 1. 4 2. From Karol A. Penson, May 0. Start)Relation between the Bell numbers B(n) and the n- th derivative of 1/Gamma(1+x) of such derivatives through seq(subs(x=0, simplify(diff(GAMMA(1+x)^(- 1),x$n))), n=1. Psi(k)) and polygamma (Psi(k,n)) functions and unevaluated; Examples of such expressions, for n=1. Psi(1),n=2: - (- Psi(1)^2+Psi(1,1)),n=3: - Psi(1)^3+3*Psi(1)*Psi(1,1)- Psi(2,1),n=4: - (- Psi(1)^4+6*Psi(1)^2*Psi(1,1)- 3*Psi(1,1)^2- 4*Psi(1)*Psi(2,1)+Psi(3, 1)),n=5: - Psi(1)^5 +1. Psi(1)^3*Psi(1,1) - 1. Psi(1)*Psi(1,1)^2 - 1. Psi(1)^2*Psi(2,1) +1. Psi(1,1)*Psi(2,1) +5*Psi(1)*Psi(3,1) - Psi(4,1); c) for a given n, read off the sum of absolute values of coefficients of every term involving digamma or polygamma functions. This sum is equal to B(n). Examples: B(1)=1, B(2)=1+1=2, B(3)=1+3+1=5, B(4)=1+6+3+4+1=1. B(5)=1+1. 0+1. 5+1. Observe that this decomposition of the Bell number B(n) apparently does not involve the Stirling numbers of the second kind explicitly.(End)The numbers given above by Penson lead to the multinomial coefficients A0. Meijer, Aug 1. 4 2. Column 1 of A1. 62. Adams- Watters, Jul 0. Asymptotic expansions (0!+1!+2!+..+(n- 1)!)/(n- 1)! Adamson, Sep 0. 6 2. A1. 65. 19. 6(2^n); where A1. Adamson, Sep 0. 6 2. The divergent series g(x=1,m) = 1^m*1! We discovered that g(x=1,m) = (- 1)^m * (A0. A0. 00. 11. 0(m+1) * A0. We observe that A0. Gompertz's constant and that A0. Gould, see for more information A1. Meijer, Oct 1. 6 2. E(X^n), i. e., the n- th moment about the origin of a random variable X that has a Poisson distribution with (rate) parameter, lambda = 1. Adamson, Feb 0. 9 2. The Bell numbers serve as the upper limit for the number of distinct homomorphic images from any given finite universal algebra. Every algebra homomorphism is determined by its kernel, which must be a congruence relation. As the number of possible congruence relations with respect to a finite universal algebra must be a subset of its possible equivalence classes (given by the Bell numbers), it follows naturally. Stephan comment see, e. W. Lang link under A0. Then B(x) is satisfied by A(x)/A(x^2) where A(x) = polcoeff A1. B(x) * B(x^2) * B(x^4) * B(x^8) * .. Adamson, Jul 0. 8 2. Consider a set of A0. Each bit that should toggle may or may not do so. E. g., for n=3, the 5 trajectories are 0,0,0,0; 0,1,0,0; 0,1,1,0; 0,0,1,0; 0,1,3,0. These correspond to equivalence relations on . We note that a labeled rooted tree is monotonic- labeled if the label of any parent vertex is greater than the label of any offspring vertex. Walsh, Nov 1. 1 2. D^n(exp(x)) evaluated at x = 0, where D is the operator (1+x)*d/dx. E. g., for n=2 there are 5 rhyme schemes of length 3 (aaa, aab, aba, abb, abc), and the 2 without repetitions are aba, abc. Munagi's result that the Bell numbers count partitions into subsets of nonconsecutive integers (see comment above dated Mar 2. Adamson, Jun 2. 1 2. Number of maps f: . Forberg, Dec 2. 6 2. This is the e. g. See the comment by Gary W. Adamson, Dec 0. 4 2. Pascal eigensequence. It is an eigensequence (the unique eigensequence with eigenvalue 1) of the matrix derived from the Pascal matrix by adding at the top the row . The binomial sum formula may be derived from the definition in terms of partitions: label any element X of a set S of N elements, and let X(k) be the number of subsets of S containing X with k elements. Since each subset has a unique coset, the number of partitions p(N) of S is given by p(N) = Sum. Chronos when mars meets venus altar records.rar. Venus and Mars (1997) DTS 5.1. Chronos is musical creation from talanted. 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