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A358694
Triangle read by rows. Coefficients of the polynomials H(n, x) = Sum_{k=0..n-1} Sum_{i=0..k} abs(Stirling1(n, n - i)) * x^(n - k) in ascending order of powers.
1
1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 24, 18, 7, 1, 0, 120, 96, 46, 11, 1, 0, 720, 600, 326, 101, 16, 1, 0, 5040, 4320, 2556, 932, 197, 22, 1, 0, 40320, 35280, 22212, 9080, 2311, 351, 29, 1, 0, 362880, 322560, 212976, 94852, 27568, 5119, 583, 37, 1
OFFSET
0,5
FORMULA
Sum_{k=0..n} T(n, k) = n! * H(n), where H(n) are the harmonic numbers (with H(0)= 1). In other words, the polynomials H(n, x) / n! evaluate at x = 1 to H(n).
EXAMPLE
[0] 1;
[1] 0, 1;
[2] 0, 2, 1;
[3] 0, 6, 4, 1;
[4] 0, 24, 18, 7, 1;
[5] 0, 120, 96, 46, 11, 1;
[6] 0, 720, 600, 326, 101, 16, 1;
[7] 0, 5040, 4320, 2556, 932, 197, 22, 1;
[8] 0, 40320, 35280, 22212, 9080, 2311, 351, 29, 1;
[9] 0, 362880, 322560, 212976, 94852, 27568, 5119, 583, 37, 1;
MAPLE
T := proc(n, k) option remember;
if k = n then 1
elif k <= 0 then 0
elif k = 1 then n*T(n-1, 1)
else T(n - 1, k - 1) + (n - 1)*T(n - 1, k) fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
# Alternative:
H := proc(n, x) if n = 0 then return 1 fi;
add(add(abs(Stirling1(n, n-i))*x^(n-k), i = 0..k), k = 0..n-1) end:
for n from 0 to 9 do seq(coeff(H(n, x), x, k), k = 0..n) od;
MATHEMATICA
T[n_, k_] := T[n, k] = Which[k == n, 1, k <= 0, 0, k == 1, n*T[n - 1, 1], True, T[n - 1, k - 1] + (n - 1)*T[n - 1, k]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 12 2023, from first Maple program *)
PROG
(Python)
from functools import cache
@cache
def A358694row(n: int) -> list[int]:
if n == 0: return [1]
if n == 1: return [0, 1]
row: list[int] = A358694row(n - 1) + [1]
sav: int = row[1]
for k in range(n - 1, 0, -1):
row[k] = (n - 1) * row[k] + row[k - 1]
row[1] += sav
return row
for n in range(10): print(A358694row(n))
CROSSREFS
Cf: A000254 (row sums), A001008 (harmonic numbers).
Sequence in context: A127631 A122538 A090238 * A047922 A276891 A021830
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Nov 27 2022
STATUS
approved