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A065931 Triangle of coefficients of Bessel polynomials {y_n(x)}'.

%I A065931 #21 Apr 05 2025 04:56:33
%S A065931 1,3,6,6,30,45,10,90,315,420,15,210,1260,3780,4725,21,420,3780,18900,
%T A065931 51975,62370,28,756,9450,69300,311850,810810,945945,36,1260,20790,
%U A065931 207900,1351350,5675670,14189175,16216200,45,1980,41580,540540,4729725,28378350,113513400,275675400,310134825
%N A065931 Triangle of coefficients of Bessel polynomials {y_n(x)}'.
%D A065931 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%H A065931 G. C. Greubel, <a href="/A065931/b065931.txt">Rows n = 1..100 of triangle, flattened</a>
%H A065931 <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>
%F A065931 From _G. C. Greubel_, Jul 10 2019: (Start)
%F A065931 (y_{n}(x))' = (1/2)*Sum_{k=0..n-1} ((n+k+1)!/(k!*(n-k-1)!))*(x/2)^k.
%F A065931 T(n, k) = ((n+k+1)!/(k!*(n-k-1)!))*(1/2)^(k+1) for 0 <= k <= n-1, n>=1. (End)
%e A065931 For n = 1 .. 4 the polynomials are
%e A065931   (y_{1}(x))' =  1;
%e A065931   (y_{2}(x))' =  3 +  6*x;
%e A065931   (y_{3}(x))' =  6 + 30*x +  45*x^2;
%e A065931   (y_{4}(x))' = 10 + 90*x + 315*x^2 + 420*x^3.
%t A065931 Table[(n+k+1)!/(k!*(n-k-1)!)*(1/2)^(k+1), {n,1,12}, {k,0,n-1}]//Flatten (* _G. C. Greubel_, Jul 10 2019 *)
%o A065931 (PARI) for(n=1,12, for(k=0,n-1, print1((n+k+1)!/(k!*(n-k-1)!)*(1/2)^(k+1), ", "))) \\ _G. C. Greubel_, Jul 10 2019
%o A065931 (Magma) f:=Factorial; [(f(n+k+1)/(f(k)*f(n-k-1)))*(1/2)^(k+1): k in [0..n-1], n in [1..12]]; // _G. C. Greubel_, Jul 10 2019
%o A065931 (Sage) f=factorial; [[(f(n+k+1)/(f(k)*f(n-k-1)))*(1/2)^(k+1) for k in (0..n-1)] for n in (1..12)] # _G. C. Greubel_, Jul 10 2019
%o A065931 (GAP) f:=Factorial;; Flat(List([1..12], n-> List([0..n-1], k-> (f(n+k+1)/(f(k)*f(n-k-1)))*(1/2)^(k+1) ))); # _G. C. Greubel_, Jul 10 2019
%Y A065931 Cf. A001497, A001498, A065943.
%K A065931 nonn,tabl
%O A065931 1,2
%A A065931 _N. J. A. Sloane_, Dec 08 2001