cp's OEIS Frontend

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.

%I A340554 #15 Dec 30 2024 02:13:28
%S A340554 1,1,1,3,1,10,5,1,36,126,84,9,1,136,2380,12376,24310,19448,6188,680,
%T A340554 17,1,528,40920,1107568,13884156,92561040,354817320,818809200,
%U A340554 1166803110,1037158320,573166440,193536720,38567100,4272048,237336,5456,33
%N 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.
%H A340554 G. C. Greubel, <a href="/A340554/b340554.txt">Rows n = 0..11 of the irregular triangle, flattened</a>
%F A340554 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
%e A340554 Triangle starts:
%e A340554                            [0] 1, 1
%e A340554                            [1] 1, 3
%e A340554                          [2] 1, 10, 5
%e A340554                      [3] 1, 36, 126, 84, 9
%e A340554      [4] 1, 136, 2380, 12376, 24310, 19448, 6188, 680, 17
%p A340554 CoeffList := p -> op(PolynomialTools:-CoefficientList(p, x)):
%p A340554 Tpoly := proc(n) simplify(hypergeom([-2^n/2, -2^n/2 - 1/2], [1/2], x)):
%p A340554 CoeffList(%) end: seq(Tpoly(n), n = 0..5);
%t A340554 Tpoly[n_] := HypergeometricPFQ[{-2^n/2, -2^n/2 - 1/2}, {1/2}, x];
%t A340554 Table[CoefficientList[Tpoly[n], x], {n, 0, 5}] // Flatten
%o A340554 (Magma)
%o A340554 p:= func< n | n eq 0 select 1 else 2^(n-1) >;
%o A340554 T:= func< n,k | Factorial(2^n+1)/(Factorial(2*k)*Factorial(2^n-2*k+1)) >;
%o A340554 [T(n,k): k in [0..p(n)], n in [0..8]]; // _G. C. Greubel_, Dec 30 2024
%o A340554 (SageMath)
%o A340554 # from sage.all import * # (use for Python)
%o A340554 def p(n): return 1 if n==0 else pow(2,n-1)
%o A340554 def T(n,k): return rising_factorial(-pow(2,n)-1, 2*k)/factorial(2*k)
%o A340554 print(flatten([[T(n,k) for k in range(p(n)+1)] for n in range(8)])) # _G. C. Greubel_, Dec 30 2024
%Y A340554 Cf. A001146 (row sums), A000051 (main diagonal), A016131 (central terms), A201461, A028297.
%K A340554 nonn,tabf
%O A340554 0,4
%A A340554 _Peter Luschny_, Feb 03 2021