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

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

Offset: 1

Author

Max Alekseyev, Nov 07 2006

Keywords

• nonn

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

Examples

			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
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000
• Eric Weisstein's World of Mathematics, Bell Polynomial
• Eric Weisstein's World of Mathematics, Faà di Bruno's Formula

Crossrefs

Cf. A000110, A000258, A000670, A005651, A008277, A008480, A056239, A094416, A124794, A215366, A318762, A319182, A319225.

Cf. A065475, A100484.

Programs

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 *)

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

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