OFFSET
0,8
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).
FORMULA
A(n, k) = Product_{j=0..n-1} (3*j + k).
A(n, k) = A(n+1, k-3) / (k - 3) for k > 3.
A(n, k) = Sum_{j=0..n} Stirling1(n, j)*(-3)^(n - j)* k^j.
A(n, k) = k! * [x^k] (exp(x) * p(n, x)), where p(n, x) are the row polynomials of A371080.
E.g.f. of column k: (1 - 3*t)^(-k/3).
E.g.f. of row n: exp(x) * (Sum_{k=0..n} A371076(n, k) * x^k / (k!)).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n / (n!) = 1/(1 - x/(1 - 3*t)^(1/3)).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n /(n! * k!) = exp(x/(1 - 3*t)^(1/3)).
EXAMPLE
The array starts:
[0] 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
[1] 0, 1, 2, 3, 4, 5, 6, 7, 8, ...
[2] 0, 4, 10, 18, 28, 40, 54, 70, 88, ...
[3] 0, 28, 80, 162, 280, 440, 648, 910, 1232, ...
[4] 0, 280, 880, 1944, 3640, 6160, 9720, 14560, 20944, ...
[5] 0, 3640, 12320, 29160, 58240, 104720, 174960, 276640, 418880, ...
.
Seen as the triangle T(n, k) = A(n - k, k):
[0] 1;
[1] 0, 1;
[2] 0, 1, 1;
[3] 0, 4, 2, 1;
[4] 0, 28, 10, 3, 1;
[5] 0, 280, 80, 18, 4, 1;
[6] 0, 3640, 880, 162, 28, 5, 1;
[7] 0, 58240, 12320, 1944, 280, 40, 6, 1;
[8] 0, 1106560, 209440, 29160, 3640, 440, 54, 7, 1;
.
Illustrating the LU decomposition of A:
/ 1 \ / 1 1 1 1 1 ... \ / 1 1 1 1 1 ... \
| 0 1 | | 1 2 3 4 ... | | 0 1 2 3 4 ... |
| 0 4 2 | * | 1 3 6 ... | = | 0 4 10 18 28 ... |
| 0 28 24 6 | | 1 4 ... | | 0 28 80 162 280 ... |
| 0 280 320 144 24 | | 1 ... | | 0 280 880 1944 3640 ... |
| . . . | | . . . | | . . . |. (End)
MAPLE
A := (n, k) -> 3^n*pochhammer(k/3, n):
A := (n, k) -> local j; mul(3*j + k, j = 0..n-1):
# Read by antidiagonals:
T := (n, k) -> A(n - k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);
seq(lprint([n], seq(T(n, k), k = 0..n)), n = 0..9);
# Using the generating polynomials of the rows:
P := (n, x) -> local k; add(Stirling1(n, k)*(-3)^(n - k)*x^k, k=0..n):
seq(lprint([n], seq(P(n, k), k = 0..9)), n = 0..5);
# Using the exponential generating functions of the columns:
EGFcol := proc(k, len) local egf, ser, n; egf := (1 - 3*x)^(-k/3);
ser := series(egf, x, len+2): seq(n!*coeff(ser, x, n), n = 0..len) end:
seq(lprint([k], EGFcol(k, 8)), k = 0..6);
# As a matrix product:
with(LinearAlgebra):
L := Matrix(7, 7, (n, k) -> A371076(n - 1, k - 1)):
U := Matrix(7, 7, (n, k) -> binomial(n - 1, k - 1)):
MatrixMatrixMultiply(L, Transpose(U));
MATHEMATICA
Table[3^(n-k)*Pochhammer[k/3, n-k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Mar 14 2024 *)
PROG
(SageMath)
def A(n, k): return 3**n * rising_factorial(k/3, n)
def A(n, k): return (-3)**n * falling_factorial(-k/3, n)
CROSSREFS
KEYWORD
AUTHOR
Werner Schulte and Peter Luschny, Mar 10 2024
STATUS
approved