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A186430 Generalized Pascal triangle associated with the set of primes. 7
1, 1, 1, 1, 2, 1, 1, 12, 12, 1, 1, 2, 12, 2, 1, 1, 120, 120, 120, 120, 1, 1, 2, 120, 20, 120, 2, 1, 1, 252, 252, 2520, 2520, 252, 252, 1, 1, 2, 252, 42, 2520, 42, 252, 2, 1, 1, 240, 240, 5040, 5040, 5040, 5040, 240, 240, 1, 1, 2, 240, 40, 5040, 84, 5040, 40, 240, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Given a subset S of the integers Z, Bhargava has shown how to associate with S a generalized factorial function, denoted n!_S, sharing many properties of the classical factorial function n! (which corresponds to the choice S = Z). In particular, he shows that the generalized binomial coefficients n!_S/(k!_S*(n-k)!_S) are always integral for any choice of S.
Here we take S = {2,3,5,7,...} the set of primes.
The generalized factorial n!_S is given by the formula n!_S = product {primes p} (p^(floor(n/(p-1)) + floor(n/(p^2-p)) + floor(n/(p^3-p^2)) + ...), and appears in the database as n!_S = A053657(n) for n>=1. We make the convention that 0!_S = 1.
See A186432 for the generalized Pascal triangle associated with the set
of squares.
LINKS
M. Bhargava, The factorial function and generalizations, Amer. Math. Monthly, 107(2000), 783-799.
FORMULA
T(n,k) = A053657(n)/(A053657(k)*A053657(n-k)), for n,k >= 0, with the convention that A053657(0) = 1.
Row sums A186431.
EXAMPLE
Triangle begins
n/k.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....1
.2..|..1.....2.....1
.3..|..1....12....12.....1
.4..|..1.....2....12.....2.....1
.5..|..1...120...120...120...120.....1
.6..|..1.....2...120....20...120.....2.....1
.7..|..1...252...252..2520..2520...252...252.....1
.8..|
MAPLE
#Uses program for A053657 written by Peter Luschny
A053657 := proc(n) local P, p, q, s, r;
P := select(isprime, [$2..n]); r:=1;
for p in P do s := 0; q := p-1;
do if q > (n-1) then break fi;
s := s + iquo(n-1, q); q := q*p; od;
r := r * p^s; od; r end:
T := (n, k) -> A053657(n)/(A053657(k)*A053657(n-k)):
for n from 0 to 10 do
seq(T(n, k), k = 0..n)
end do;
MATHEMATICA
b[n_] := Product[p^Sum[Floor[(n - 1)/((p - 1) p^k)], {k, 0, n}], {p, Prime[ Range[n]]}];
T[n_, k_] := b[n]/(b[k] b[n - k]);
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 22 2019 *)
CROSSREFS
Sequence in context: A324188 A297762 A010246 * A173889 A156885 A174718
KEYWORD
nonn,easy,tabl
AUTHOR
Peter Bala, Feb 21 2011
STATUS
approved

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Last modified February 22 19:53 EST 2024. Contains 370260 sequences. (Running on oeis4.)