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A124794 Coefficients of incomplete Bell polynomials in the prime factorization order. 66
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
COMMENTS
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
LINKS
Eric Weisstein's World of Mathematics, Bell Polynomial
Eric Weisstein's World of Mathematics, Faà di Bruno's Formula
FORMULA
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*...)).
a(2*prime(n)) = n + 1, for n > 1. See A065475. - Bill McEachen, Oct 11 2023
EXAMPLE
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
MAPLE
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))))]):
seq(a(n), n=1..100); # Alois P. Heinz, Feb 14 2020
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Sequence in context: A260419 A117184 A035690 * A206496 A097560 A218905
KEYWORD
nonn
AUTHOR
Max Alekseyev, Nov 07 2006
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)