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, 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

Links

Table of n, a(n) for n=0..54.

Peter Luschny, Illustrating the polynomials.

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

Adjacent sequences: A358691 A358692 A358693 * A358695 A358696 A358697

Keyword

nonn,tabl

Author

Peter Luschny, Nov 27 2022

Status

approved