OFFSET
1,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 185 terms from R. H. Hardin)
FORMULA
T(n,k) = Sum_{d|n} phi(n/d) * A201552(d, k). - Andrew Howroyd, Oct 14 2017
Empirical for row n:
n=1: a(k) = 1.
n=2: a(k) = k + 1.
n=3: a(k) = k^2 + k + 1.
n=4: a(k) = (4/3)*k^3 + 2*k^2 + (5/3)*k + 1.
n=5: a(k) = (23/12)*k^4 + (23/6)*k^3 + (37/12)*k^2 + (7/6)*k + 1.
n=6: a(k) = (44/15)*k^5 + (22/3)*k^4 + (23/3)*k^3 + (14/3)*k^2 + (12/5)*k + 1.
n=7: a(k) = (841/180)*k^6 + (841/60)*k^5 + (325/18)*k^4 + (51/4)*k^3 + (949/180)*k^2 + (37/30)*k + 1.
EXAMPLE
Table starts
...1....1.....1......1.......1.......1........1........1........1.........1
...2....3.....4......5.......6.......7........8........9.......10........11
...3....7....13.....21......31......43.......57.......73.......91.......111
...6...23....60....125.....226.....371......568......825.....1150......1551
..11...77...291....791....1761....3431.....6077....10021....15631.....23321
..26..297..1564...5457...14838...34153....69784...130401...227314....374825
..57.1163..8671..39019..129823..353333...833253..1764925..3438877...6267735
.142.4783.49852.288317.1172298.3770475.10259448.24627705.53630854.108036775
MATHEMATICA
comps[r_, m_, k_] := Sum[(-1)^i*Binomial[r-1-i*m, k-1]*Binomial[k, i], {i, 0, Floor[(r-k)/m]}]; a[n_, k_] := DivisorSum[n, EulerPhi[n/#] comps[#*(k + 1), 2k+1, #]&]/n; Table[a[n-k+1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 07 2017, after Andrew Howroyd's PARI code *)
PROG
(PARI)
comps(r, m, k)=sum(i=0, floor((r-k)/m), (-1)^i*binomial(r-1-i*m, k-1)*binomial(k, i));
a(n, k)=sumdiv(n, d, eulerphi(n/d)*comps(d*(k+1), 2*k+1, d))/n;
for(n=1, 8, for(k=1, 10, print1(a(n, k), ", ")); print()); \\ Andrew Howroyd, May 16 2017
(Python)
from sympy import binomial, divisors, totient, floor
def comps(r, m, k): return sum([(-1)**i*binomial(r - 1 - i*m, k - 1)*binomial(k, i) for i in range(floor((r - k)/m) + 1)])
def a(n, k): return sum([totient(n//d)*comps(d*(k + 1), 2*k + 1, d) for d in divisors(n)])//n
for n in range(1, 12): print([a(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Nov 07 2017, after PARI code
(R)
require(numbers)
comps <- function(r, m, k) {
S <- numeric()
for (i in 0:floor((r-k)/m)) S <- c(S, (-1)^i*choose(r-1-i*m, k-1)*choose(k, i))
return(sum(S))
}
a <- function(n, k) {
S <- numeric()
for (d in divisors(n)) S <- c(S, eulersPhi(n/d)*comps(d*(k+1), 2*k+1, d))
return(sum(S)/n)
}
for (n in 1:11) {
for (k in 1:n) {
print(a(k, n-k+1))
}
} # Indranil Ghosh, Nov 07 2017, after PARI code
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Feb 29 2012
STATUS
approved