A067147 Triangle of coefficients for expressing x^n in terms of Hermite polynomials.
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
Offset: 0
Examples
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;
References
- 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)
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
- 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].
- M. Griffin, K. Ono, L. Rolen, and D. Zagier, Jensen polynomials for the Riemann zeta function and other sequences, arXiv:1902.07321 [math.NT], 2019.
- Index entries for sequences related to Hermite polynomials
Crossrefs
Programs
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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
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Maple
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
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 *) -
PARI
T(n, k) = round(n!*(1+(-1)^(n+k))/(2*k! *gamma((n-k+2)/2))) for(n=0,20, for(k=0,n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jun 09 2018
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PARI
{T(n,k) = if(k<0 || nMichael Somos, Jan 15 2020 */
Formula
E.g.f. (rel to x): A(x, y) = exp(x*y + x^2).
Sum_{ k>=0 } 2^k*k!*T(m, k)*T(n, k) = T(m+n, 0) = |A067994(m+n)|. - Philippe Deléham, Jul 02 2005
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
Comments