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A124794 Coefficients of incomplete Bell polynomials in the prime factorization order. 65
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

Alois P. Heinz, Table of n, a(n) for n = 1..20000

MathWorld, Bell Polynomial

MathWorld, FaĆ  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*...).

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 *)

CROSSREFS

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

Sequence in context: A260419 A117184 A035690 * A206496 A097560 A218905

Adjacent sequences:  A124791 A124792 A124793 * A124795 A124796 A124797

KEYWORD

nonn

AUTHOR

Max Alekseyev, Nov 07 2006

STATUS

approved

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Last modified July 6 00:43 EDT 2020. Contains 335475 sequences. (Running on oeis4.)