OFFSET
0,8
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Bessel Polynomial
FORMULA
From G. C. Greubel, Jan 02 2023: (Start)
T(n, k) = binomial(n-1,k)*(n+k-1)!/(2^k*(n-1)!), with T(n, n) = 0.
Sum_{k=0..n} T(n, k) = A001515(n-1).
Sum_{k=0..n} (-1)^k*T(n, k) = A000806(n-1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000085(n-1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A001464(n-1). (End)
EXAMPLE
Bessel polynomials begin with:
x;
x + x^2;
3*x + 3*x^2 + x^3;
15*x + 15*x^2 + 6*x^3 + x^4;
105*x + 105*x^2 + 45*x^3 + 10*x^4 + x^5;
...
Triangle of coefficients begins as:
0;
1, 0;
1, 1 0;
1, 3, 3 0;
1, 6, 15, 15 0;
1, 10, 45, 105, 105 0;
1, 15, 105, 420, 945, 945 0;
1, 21, 210, 1260, 4725, 10395, 10395 0;
1, 28, 378, 3150, 17325, 62370, 135135, 135135 0;
MATHEMATICA
T[n_, k_]:= If[k==n, 0, Binomial[n-1, k]*(n+k-1)!/(2^k*(n-1)!)];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 02 2023 *)
PROG
(Magma)
A104548:= func< n, k | k eq n select 0 else Binomial(n-1, k)*Factorial(n+k-1)/(2^k*Factorial(n-1)) >;
[A104548(n, k): k in [0..n], n in [0..13]]; // G. C. Greubel, Jan 02 2023
(SageMath)
def A104548(n, k): return 0 if (k==n) else binomial(n-1, k)*factorial(n+k-1)/(2^k*factorial(n-1))
flatten([[A104548(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 02 2023
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Eric W. Weisstein, Mar 14 2005
EXTENSIONS
T(0, 0) = 0 prepended by G. C. Greubel, Jan 02 2023
STATUS
approved