OFFSET
0,4
LINKS
G. C. Greubel, Rows n = 0..11 of the irregular triangle, flattened
FORMULA
T(n, k) = (2^n + 1)!/((2*k)! * (2^n - 2*k + 1)!), for n >= 0, 0 <= k <= p(n), where p(n) = 1 if n = 0 otherwise p(n) = 2^(n-1). Alternative form: T(n, k) = Pochhammer(-2^n - 1, 2*k)/(2*k)!. - G. C. Greubel, Dec 30 2024
EXAMPLE
Triangle starts:
[0] 1, 1
[1] 1, 3
[2] 1, 10, 5
[3] 1, 36, 126, 84, 9
[4] 1, 136, 2380, 12376, 24310, 19448, 6188, 680, 17
MAPLE
CoeffList := p -> op(PolynomialTools:-CoefficientList(p, x)):
Tpoly := proc(n) simplify(hypergeom([-2^n/2, -2^n/2 - 1/2], [1/2], x)):
CoeffList(%) end: seq(Tpoly(n), n = 0..5);
MATHEMATICA
Tpoly[n_] := HypergeometricPFQ[{-2^n/2, -2^n/2 - 1/2}, {1/2}, x];
Table[CoefficientList[Tpoly[n], x], {n, 0, 5}] // Flatten
PROG
(Magma)
p:= func< n | n eq 0 select 1 else 2^(n-1) >;
T:= func< n, k | Factorial(2^n+1)/(Factorial(2*k)*Factorial(2^n-2*k+1)) >;
[T(n, k): k in [0..p(n)], n in [0..8]]; // G. C. Greubel, Dec 30 2024
(SageMath)
# from sage.all import * # (use for Python)
def p(n): return 1 if n==0 else pow(2, n-1)
def T(n, k): return rising_factorial(-pow(2, n)-1, 2*k)/factorial(2*k)
print(flatten([[T(n, k) for k in range(p(n)+1)] for n in range(8)])) # G. C. Greubel, Dec 30 2024
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Feb 03 2021
STATUS
approved