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

%I A067147 #53 Aug 20 2024 15:31:46
%S A067147 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 A067147 840,0,420,0,42,0,1,1680,0,3360,0,840,0,56,0,1,0,15120,0,10080,0,1512,
%U A067147 0,72,0,1,30240,0,75600,0,25200,0,2520,0,90,0,1
%N A067147 Triangle of coefficients for expressing x^n in terms of Hermite polynomials.
%C A067147 x^n = (1/2^n) * Sum_{k=0..n} a(n,k)*H_k(x).
%C A067147 These polynomials, H_n(x), are an Appell sequence, whose umbral compositional inverse sequence HI_n(x) consists of the same polynomials signed with the e.g.f. e^{-t^2} e^{xt}. Consequently, under umbral composition H_n(HI.(x)) = x^n = HI_n(H.(x)). Other differently scaled families of Hermite polynomials are A066325, A099174, and A060821. See Griffin et al. for a relation to the Catalan numbers and matrix integration. - _Tom Copeland_, Dec 27 2020
%D A067147 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 A067147 G. C. Greubel, <a href="/A067147/b067147.txt">Rows n = 0..100 of triangle, flattened</a>
%H A067147 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A067147 M. Griffin, K. Ono, L. Rolen, and D. Zagier, <a href="https://arxiv.org/abs/1902.07321">Jensen polynomials for the Riemann zeta function and other sequences</a>, arXiv:1902.07321 [math.NT], 2019.
%H A067147 <a href="/index/He#Hermite">Index entries for sequences related to Hermite polynomials</a>
%F A067147 E.g.f. (rel to x): A(x, y) = exp(x*y + x^2).
%F A067147 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 A067147 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 A067147 T(n, k) = n!/(k!*2^((n-k)/2)*((n-k)/2)!)*2^((n+k)/2)*(1+(-1)^(n+k))/2^(k+1).
%F A067147 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 A067147 Exponential Riordan array (e^(x^2),x). - _Paul Barry_, Sep 12 2006
%F A067147 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). - _Paul Barry_, Apr 10 2009
%F A067147 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
%F A067147 As noted in the comments this is an Appell sequence of polynomials, so the lowering and raising operators defined by L H_n(x) = n H_{n-1}(x) and R H_{n}(x) = H_{n+1}(x) are L = D_x, the derivative, and R = D_t log[e^{t^2} e^{xt}] |_{t = D_x} = x + 2 D_x, and the polynomials may also be generated by e^{-D^2} x^n = H_n(x). - _Tom Copeland_, Dec 27 2020
%e A067147 Triangle begins with:
%e A067147     1;
%e A067147     0,   1;
%e A067147     2,   0,   1;
%e A067147     0,   6,   0,   1;
%e A067147    12,   0,  12,   0,   1;
%e A067147     0,  60,   0,  20,   0,   1;
%e A067147   120,   0, 180,   0,  30,   0,   1;
%p A067147 T := proc(n, k) (n - k)/2; `if`(%::integer, (n!/k!)/%!, 0) end:
%p A067147 for n from 0 to 11 do seq(T(n, k), k=0..n) od; # _Peter Luschny_, Jan 05 2021
%t A067147 Table[n!*(1+(-1)^(n+k))/(2*k!*Gamma[(n-k+2)/2]), {n,0,20}, {k,0,n}]// Flatten (* _G. C. Greubel_, Jun 09 2018 *)
%o A067147 (PARI) T(n, k) = round(n!*(1+(-1)^(n+k))/(2*k! *gamma((n-k+2)/2)))
%o A067147 for(n=0,20, for(k=0,n, print1(T(n, k), ", "))) \\ _G. C. Greubel_, Jun 09 2018
%o A067147 (PARI) {T(n,k) = if(k<0 || n<k || (n-k)%2, 0, n!/(k!*((n-k)/2)!))}; /* _Michael Somos_, Jan 15 2020 */
%o A067147 (Magma) [[Round(Factorial(n)*(1+(-1)^(n+k))/(2*Factorial(k)*Gamma((n-k+2)/2))): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Jun 09 2018
%Y A067147 Row sums give A047974. Columns 0-2: A001813, A000407, A001814. Cf. A048854, A060821.
%Y A067147 Cf. A060821, A066325, and A099174.
%K A067147 nonn,tabl
%O A067147 0,4
%A A067147 _Christian G. Bower_, Jan 03 2002