|
| |
|
|
A346083
|
|
Triangle, read by rows, defined by recurrence: T(n,k) = T(n-1,k-1) + (-1)^k * (2 * k + 1) * T(n-1,k) for 0 < k < n with initial values T(n,0) = T(n,n) = 1 for n >= 0 and T(i,j) = 0 if j < 0 or j > i.
|
|
0
|
|
|
|
1, 1, 1, 1, -2, 1, 1, 7, 3, 1, 1, -20, 22, -4, 1, 1, 61, 90, 50, 5, 1, 1, -182, 511, -260, 95, -6, 1, 1, 547, 2373, 2331, 595, 161, 7, 1, 1, -1640, 12412, -13944, 7686, -1176, 252, -8, 1, 1, 4921, 60420, 110020, 55230, 20622, 2100, 372, 9, 1, 1, -14762, 307021, -709720, 607090, -171612, 47922, -3480, 525, -10, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. of column k >= 0: col(t,k) = Sum_{n >= k} T(n,k) * t^n = t^k / (Product_{i=0..k} (1 - (-1)^i * (2 * i + 1) * t)), i.e., col(t,k) = col(t,k-1) * t / (1 - (-1)^k * (2 * k + 1) * t) for k > 0.
Matrix inverse M = T^(-1) has row polynomials p(n,x) = Sum_{k=0..n} M(n,k) * x^k = Product_{i=1..n} (x + (-1)^i * (2 * i - 1)) for n >= 0 and empty product 1, i.e., p(n,x) = p(n-1,x) * (x + (-1)^n * (2 * n - 1)) for n > 0 with initial value p(0,x) = 1.
Conjecture: E.g.f. of column k >= 0: Sum_{n >= k} T(n,k) * t^n / (n!) = (Sum_{i=0..k} (-1)^(i * (i + 1 ) / 2) * binomial(k,floor((k - i) / 2)) * exp((-1)^i * (2 * i + 1) * t)) * (-1)^(k * (k - 1) / 2) / (4^k * (k!)), i.e., T(n,k) = (Sum_{i=0..k} (-1)^(i * (i + 1) / 2) * binomial(k,floor((k - i) / 2)) * ((-1)^i * (2 * i + 1))^n) * (-1)^(k * (k - 1) / 2) / (4^k * (k!)) for 0 <= k <= n.
Conjecture: E.g.f. of column k >= 0: Sum_{n >= k} T(n,k) * t^n / (n!) = exp(t) * (exp(4*t) - 1)^k / (4^k * (k!) * exp(4*t*floor((k+1)/2))), i.e., T(n,k) = (Sum_{i=0..k} (-1)^i * binomial(k,i) *(1 + 4*i - 4*floor((k+1)/2))^n) * (-1)^k / (4^k * (k!)) for 0 <= k <= n. Proved by Burkhard Hackmann and Werner Schulte (distinction of two cases: odd k, even k). - Werner Schulte, Aug 03 2021
|
|
|
EXAMPLE
|
The triangle T(n,k) for 0 <= k <= n starts:
n\k : 0 1 2 3 4 5 6 7 8 9
=============================================================
0 : 1
1 : 1 1
2 : 1 -2 1
3 : 1 7 3 1
4 : 1 -20 22 -4 1
5 : 1 61 90 50 5 1
6 : 1 -182 511 -260 95 -6 1
7 : 1 547 2373 2331 595 161 7 1
8 : 1 -1640 12412 -13944 7686 -1176 252 -8 1
9 : 1 4921 60420 110020 55230 20622 2100 372 9 1
etc.
|
|
|
PROG
|
(Python)
from functools import cache
@cache
def T(n, k):
if k == 0 or k == n: return 1
return T(n-1, k-1) + (-1)**k*(2*k + 1)*T(n-1, k)
for n in range(10):
print([T(n, k) for k in range(n+1)]) # Peter Luschny, Jul 22 2021
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
