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A124794
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Coefficients of incomplete Bell polynomials in the prime factorization order.
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66
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1, 1, 1, 1, 1, 3, 1, 1, 3, 4, 1, 6, 1, 5, 10, 1, 1, 15, 1, 10, 15, 6, 1, 10, 10, 7, 15, 15, 1, 60, 1, 1, 21, 8, 35, 45, 1, 9, 28, 20, 1, 105, 1, 21, 105, 10, 1, 15, 35, 70, 36, 28, 1, 105, 56, 35, 45, 11, 1, 210, 1, 12, 210, 1, 84, 168, 1, 36, 55, 280, 1, 105, 1, 13, 280, 45, 126, 252, 1
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OFFSET
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1,6
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COMMENTS
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Coefficients of (D^k f)(g(t))*(D g(t))^k1*(D^2 g(t))^k2*... in the Faa di Bruno formula for D^m(f(g(t))) where k = k1 + k2 + ..., m = 1*k1 + 2*k2 + ....
Number of set partitions whose block sizes are the prime indices of n (i.e., the integer partition with Heinz number n). - Gus Wiseman, Sep 12 2018
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LINKS
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FORMULA
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For n = p1^k1*p2^k2*... where 2 = p1 < p2 < ... are the sequence of all primes, a(n) = a([k1,k2,...]) = (k1+2*k2+...)!/((k1!*k2!*...)*(1!^k1*2!^k2*...)).
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EXAMPLE
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The a(6) = 3 set partitions of type (2,1) are {{1},{2,3}}, {{1,3},{2}}, {{1,2},{3}}. - Gus Wiseman, Sep 12 2018
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MAPLE
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with(numtheory):
a:= n-> (l-> add(i*l[i], i=1..nops(l))!/mul(l[i]!*i!^l[i],
i=1..nops(l)))([seq(padic[ordp](n, ithprime(i)),
i=1..pi(max(1, factorset(n))))]):
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MATHEMATICA
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numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[numSetPtnsOfType[If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]], {n, 100}] (* Gus Wiseman, Sep 12 2018 *)
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PROG
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(PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, primepi(f[k, 1])*f[k, 2])!/(prod(k=1, #f~, f[k, 2]!)*prod(k=1, #f~, primepi(f[k, 1])!^f[k, 2])); \\ Michel Marcus, Oct 11 2023
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CROSSREFS
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Cf. A000110, A000258, A000670, A005651, A008277, A008480, A056239, A094416, A124794, A215366, A318762, A319182, A319225.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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