A340554 T(n, k) = [x^k] hypergeom([-2^n/2, -2^n/2 - 1/2], [1/2], x). Triangle read by rows, T(n, k) for n >= 0.

1, 1, 1, 3, 1, 10, 5, 1, 36, 126, 84, 9, 1, 136, 2380, 12376, 24310, 19448, 6188, 680, 17, 1, 528, 40920, 1107568, 13884156, 92561040, 354817320, 818809200, 1166803110, 1037158320, 573166440, 193536720, 38567100, 4272048, 237336, 5456, 33

Offset: 0

Author

Peter Luschny, Feb 03 2021

Keywords

• nonn
• tabf

Examples

			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
		

Links

• G. C. Greubel, Rows n = 0..11 of the irregular triangle, flattened

Crossrefs

Cf. A001146 (row sums), A000051 (main diagonal), A016131 (central terms), A201461, A028297.

Programs

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

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

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

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