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

%I A104548 #13 Feb 16 2025 08:32:57
%S A104548 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,
%T A104548 945,0,1,21,210,1260,4725,10395,10395,0,1,28,378,3150,17325,62370,
%U A104548 135135,135135,0,1,36,630,6930,51975,270270,945945,2027025,2027025,0
%N A104548 Triangle read by rows giving coefficients of Bessel polynomial p_n(x).
%H A104548 G. C. Greubel, <a href="/A104548/b104548.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A104548 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BesselPolynomial.html">Bessel Polynomial</a>
%F A104548 From _G. C. Greubel_, Jan 02 2023: (Start)
%F A104548 T(n, k) = binomial(n-1,k)*(n+k-1)!/(2^k*(n-1)!), with T(n, n) = 0.
%F A104548 Sum_{k=0..n} T(n, k) = A001515(n-1).
%F A104548 Sum_{k=0..n} (-1)^k*T(n, k) = A000806(n-1).
%F A104548 Sum_{k=0..floor(n/2)} T(n-k, k) = A000085(n-1).
%F A104548 Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A001464(n-1). (End)
%e A104548 Bessel polynomials begin with:
%e A104548       x;
%e A104548       x +     x^2;
%e A104548     3*x +   3*x^2 +    x^3;
%e A104548    15*x +  15*x^2 +  6*x^3 +    x^4;
%e A104548   105*x + 105*x^2 + 45*x^3 + 10*x^4 + x^5;
%e A104548   ...
%e A104548 Triangle of coefficients begins as:
%e A104548   0;
%e A104548   1,  0;
%e A104548   1,  1    0;
%e A104548   1,  3,   3     0;
%e A104548   1,  6,  15,   15      0;
%e A104548   1, 10,  45,  105,   105      0;
%e A104548   1, 15, 105,  420,   945,   945       0;
%e A104548   1, 21, 210, 1260,  4725, 10395,  10395       0;
%e A104548   1, 28, 378, 3150, 17325, 62370, 135135, 135135    0;
%t A104548 T[n_, k_]:= If[k==n, 0, Binomial[n-1,k]*(n+k-1)!/(2^k*(n-1)!)];
%t A104548 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 02 2023 *)
%o A104548 (Magma)
%o A104548 A104548:= func< n,k | k eq n select 0 else Binomial(n-1,k)*Factorial(n+k-1)/(2^k*Factorial(n-1)) >;
%o A104548 [A104548(n,k): k in [0..n], n in [0..13]]; // _G. C. Greubel_, Jan 02 2023
%o A104548 (SageMath)
%o A104548 def A104548(n,k): return 0 if (k==n) else binomial(n-1,k)*factorial(n+k-1)/(2^k*factorial(n-1))
%o A104548 flatten([[A104548(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jan 02 2023
%Y A104548 Essentially the same as A001498 (the main entry).
%Y A104548 Cf. A000085, A000806, A001464, A001515.
%K A104548 nonn,tabl
%O A104548 0,8
%A A104548 _Eric W. Weisstein_, Mar 14 2005
%E A104548 T(0, 0) = 0 prepended by _G. C. Greubel_, Jan 02 2023