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.

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

Offset

0,5

Links

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

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

Cf. A000012 (column 0 and main diagonal), A014983 (column 1), A181983 (1st subdiagonal), A002412 (2nd subdiagonal), A264851 (3rd subdiagonal without signs).

Cf. A051159.

Sequence in context: A344440 A259175 A297431 * A301922 A144510 A143670

Adjacent sequences: A346080 A346081 A346082 * A346084 A346085 A346086

Keyword

sign,easy,tabl

Author

Werner Schulte, Jul 04 2021

Status

approved