A065931 Triangle of coefficients of Bessel polynomials {y_n(x)}'.
1, 3, 6, 6, 30, 45, 10, 90, 315, 420, 15, 210, 1260, 3780, 4725, 21, 420, 3780, 18900, 51975, 62370, 28, 756, 9450, 69300, 311850, 810810, 945945, 36, 1260, 20790, 207900, 1351350, 5675670, 14189175, 16216200, 45, 1980, 41580, 540540, 4729725, 28378350, 113513400, 275675400, 310134825
Offset: 1
Examples
For n = 1 .. 4 the polynomials are
(y_{1}(x))' = 1;
(y_{2}(x))' = 3 + 6*x;
(y_{3}(x))' = 6 + 30*x + 45*x^2;
(y_{4}(x))' = 10 + 90*x + 315*x^2 + 420*x^3.
References
- J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
Links
Programs
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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
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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
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Mathematica
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 *) -
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
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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
Formula
From G. C. Greubel, Jul 10 2019: (Start)
(y_{n}(x))' = (1/2)*Sum_{k=0..n-1} ((n+k+1)!/(k!*(n-k-1)!))*(x/2)^k.
T(n, k) = ((n+k+1)!/(k!*(n-k-1)!))*(1/2)^(k+1) for 0 <= k <= n-1, n>=1. (End)