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A124794 Coefficients of incomplete Bell polynomials in the prime factorization order.

%I A124794 #35 Feb 16 2025 08:33:03
%S A124794 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,
%T A124794 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,
%U A124794 35,45,11,1,210,1,12,210,1,84,168,1,36,55,280,1,105,1,13,280,45,126,252,1
%N A124794 Coefficients of incomplete Bell polynomials in the prime factorization order.
%C A124794 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 + ....
%C A124794 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
%H A124794 Alois P. Heinz, <a href="/A124794/b124794.txt">Table of n, a(n) for n = 1..20000</a>
%H A124794 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>
%H A124794 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FaadiBrunosFormula.html">Faà di Bruno's Formula</a>
%F A124794 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*...)).
%F A124794 a(2*prime(n)) = n + 1, for n > 1. See A065475. - _Bill McEachen_, Oct 11 2023
%e A124794 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
%p A124794 with(numtheory):
%p A124794 a:= n-> (l-> add(i*l[i], i=1..nops(l))!/mul(l[i]!*i!^l[i],
%p A124794          i=1..nops(l)))([seq(padic[ordp](n, ithprime(i)),
%p A124794          i=1..pi(max(1, factorset(n))))]):
%p A124794 seq(a(n), n=1..100);  # _Alois P. Heinz_, Feb 14 2020
%t A124794 numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
%t A124794 Table[numSetPtnsOfType[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,100}] (* _Gus Wiseman_, Sep 12 2018 *)
%o A124794 (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
%Y A124794 Cf. A000110, A000258, A000670, A005651, A008277, A008480, A056239, A094416, A124794, A215366, A318762, A319182, A319225.
%Y A124794 Cf. A065475, A100484.
%K A124794 nonn
%O A124794 1,6
%A A124794 _Max Alekseyev_, Nov 07 2006