OFFSET
0,8
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).
FORMULA
The polynomials P(n, x) = Sum_{k=0..n} Stirling1(n, k)*(-2)^(n-k)*x^k are ordinary generating functions for row n, i.e., A(n, k) = P(n, k).
From Werner Schulte, Mar 06 and 07 2024: (Start)
A(n, k) = Product_{i=1..n} (2*i - 2 + k).
E.g.f. of column k: Sum_{n>=0} A(n, k) * t^n / (n!) = (1/sqrt(1 - 2*t))^k.
A(n, k) = A(n+1, k-2) / (k - 2) for k > 2.
A(n, k) = Sum_{i=0..k-1} i! * A265649(n, i) * binomial(k-1, i) for k > 0.
E.g.f. of row n > 0: Sum_{k>=1} A(n, k) * x^k / (k!) = (Sum_{k=1..n} A035342(n, k) * x^k) * exp(x).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n / (k! * n!) = exp(x/sqrt(1 - 2*t)).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n / (n!) = 1 / (1 - x/sqrt(1 - 2*t)).
The LU decomposition of this array is given by the upper triangular matrix U which is the transpose of A007318 and the lower triangular matrix L, where L is defined L(n, k) = A035342(n, k) * k! for 1 <= k <= n and L(n, 0) = 0^n. Note that L(n, k) + L(n, k+1) = A265649(n, k) * k! for 0 <= k <= n. (End)
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, 3, 8, 15, 24, 35, 48, 63, 80, ...
[3] 0, 15, 48, 105, 192, 315, 480, 693, 960, ...
[4] 0, 105, 384, 945, 1920, 3465, 5760, 9009, 13440, ...
[5] 0, 945, 3840, 10395, 23040, 45045, 80640, 135135, 215040, ...
.
Seen as the triangle T(n, k) = A(n - k, k):
[0] 1;
[1] 0, 1;
[2] 0, 1, 1;
[3] 0, 3, 2, 1;
[4] 0, 15, 8, 3, 1;
[5] 0, 105, 48, 15, 4, 1;
[6] 0, 945, 384, 105, 24, 5, 1;
.
From Werner Schulte, Mar 07 2024: (Start)
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 3 2 | * | 1 3 6 ... | = | 0 3 8 15 24 ... |
| 0 15 18 6 | | 1 4 ... | | 0 15 48 105 192 ... |
| 0 105 174 108 24 | | 1 ... | | 0 105 384 945 1920 ... |
| . . . | | . . . | | . . . |. (End)
MAPLE
A := (n, k) -> 2^n*pochhammer(k/2, n):
for n from 0 to 5 do seq(A(n, k), k = 0..9) od;
T := (n, k) -> A(n - k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);
# Using the exponential generating functions of the columns:
EGFcol := proc(k, len) local egf, ser, n; egf := (1 - 2*x)^(-k/2);
ser := series(egf, x, len+2): seq(n!*coeff(ser, x, n), n = 0..len) end:
seq(lprint(EGFcol(n, 9)), n = 0..8);
# Using the generating polynomials for the rows:
P := (n, x) -> local k; add(Stirling1(n, k)*(-2)^(n - k)*x^k, k=0..n):
seq(lprint([n], seq(P(n, k), k = 0..8)), n = 0..5);
# Implementing the comment of Werner Schulte about the LU decomposition of A:
with(LinearAlgebra):
L := Matrix(7, 7, (n, k) -> A371025(n - 1, k - 1)):
U := Matrix(7, 7, (n, k) -> binomial(n - 1, k - 1)):
MatrixMatrixMultiply(L, Transpose(U)); # Peter Luschny, Mar 08 2024
MATHEMATICA
A370419[n_, k_] := 2^n*Pochhammer[k/2, n];
Table[A370419[n-k, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Mar 06 2024 *)
PROG
(SageMath)
def A(n, k): return 2**n * rising_factorial(k/2, n)
for n in range(6): print([A(n, k) for k in range(9)])
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Mar 04 2024
STATUS
approved