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 A067147 Triangle of coefficients for expressing x^n in terms of Hermite polynomials. 5

%I

%S 1,0,1,2,0,1,0,6,0,1,12,0,12,0,1,0,60,0,20,0,1,120,0,180,0,30,0,1,0,

%T 840,0,420,0,42,0,1,1680,0,3360,0,840,0,56,0,1,0,15120,0,10080,0,1512,

%U 0,72,0,1,30240,0,75600,0,25200,0,2520,0,90,0,1,0,332640,0,277200,0

%N Triangle of coefficients for expressing x^n in terms of Hermite polynomials.

%C x^n = 1/2^n * Sum (a(n,k)*H_k(x)), k=0..n

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 801. (Table 22.12)

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H <a href="/index/He#Hermite">Index entries for sequences related to Hermite polynomials</a>

%F E.g.f. (rel to x) A(x, y) = exp(x*y + x^2).

%F Sum_{ k>=0 } 2^k*k!*T(m, k)*T(n, k) = T(m+n, 0) = |A067994(m+n)| . - _Philippe Deléham_, Jul 02 2005

%F T(n, k) = 0 if n-k is odd; T(n, k) = n!/(k!*((n-k)/2)!) if n-k is even . - _Philippe Deléham_, Jul 02 2005

%F T(n, k)=n!/(k!*2^((n-k)/2)((n-k)/2)!)*2^((n+k)/2)(1+(-1)^(n+k))/2^(k+1) T(n, k)=A001498((n+k)/2, (n-k)/2)2^((n+k)/2)(1+(-1)^(n+k))/2^(k+1); - _Paul Barry_, Aug 28 2005

%F Exponential Riordan array (e^(x^2),x). - _Paul Barry_, Sep 12 2006

%F G.f.: 1/(1-x*y-2*x^2/(1-x*y-4*x^2/(1-x*y-6*x^2/(1-x*y-8*x^2/(1-... (continued fraction). [From _Paul Barry_, Apr 10 2009]

%F The n-th row entries may be obtained from D^n(exp(x*t)) evaluated at x = 0, where D is the operator sqrt(1+4*x)*d/dx. - Peter Bala, Dec 07 2011

%e 1; 0,1; 2,0,1; 0,6,0,1; 12,0,12,0,1; ...

%Y Row sums give A047974. Columns 0-2: A001813, A000407, A001814. Cf. A048854, A060821.

%K nonn,tabl

%O 0,4

%A _Christian G. Bower_, Jan 03 2002

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