A104548 Triangle read by rows giving coefficients of Bessel polynomial p_n(x).

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, 1, 36, 630, 6930, 51975, 270270, 945945, 2027025, 2027025, 0

Offset: 0

Author

Eric W. Weisstein, Mar 14 2005

Keywords

• nonn
• tabl

Examples

			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;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• Eric Weisstein's World of Mathematics, Bessel Polynomial

Crossrefs

Essentially the same as A001498 (the main entry).

Cf. A000085, A000806, A001464, A001515.

Programs

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

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 *)

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

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)

Extensions

T(0, 0) = 0 prepended by G. C. Greubel, Jan 02 2023