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A067147
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Triangle of coefficients for expressing x^n in terms of Hermite polynomials.
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7
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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, 840, 0, 420, 0, 42, 0, 1, 1680, 0, 3360, 0, 840, 0, 56, 0, 1, 0, 15120, 0, 10080, 0, 1512, 0, 72, 0, 1, 30240, 0, 75600, 0, 25200, 0, 2520, 0, 90, 0, 1
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OFFSET
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0,4
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COMMENTS
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x^n = (1/2^n) * Sum_{k=0..n} a(n,k)*H_k(x).
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
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REFERENCES
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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)
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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E.g.f. (rel to x): A(x, y) = exp(x*y + x^2).
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
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
Exponential Riordan array (e^(x^2),x). - Paul Barry, Sep 12 2006
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
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
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
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EXAMPLE
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Triangle begins with:
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;
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MAPLE
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T := proc(n, k) (n - k)/2; `if`(%::integer, (n!/k!)/%!, 0) end:
for n from 0 to 11 do seq(T(n, k), k=0..n) od; # Peter Luschny, Jan 05 2021
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MATHEMATICA
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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 *)
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PROG
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(PARI) for(n=0, 20, for(k=0, n, print1(round(n!*(1+(-1)^(n+k))/(2*k! *gamma((n-k+2)/2)), ", "))) \\ G. C. Greubel, Jun 09 2018
(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
(PARI) {T(n, k) = if(k<0 || n<k || (n-k)%2, 0, n!/(k!*((n-k)/2)!))}; /* Michael Somos, Jan 15 2020 */
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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