A186430 Generalized Pascal triangle associated with the set of primes.

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

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

Table of n, a(n) for n=0..65.

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

#A186430
#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

Cf. A053657, A186431, A186432.

Sequence in context: A324188 A297762 A010246 * A173889 A156885 A174718

Adjacent sequences: A186427 A186428 A186429 * A186431 A186432 A186433

Keyword

nonn,easy,tabl

Author

Peter Bala, Feb 21 2011

Status

approved