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Search: id:A060821
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| A060821 |
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Triangle T(n,k) read by rows giving coefficients of Hermite polynomial of order n (n >= 0, 0 <= k <= n). |
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+0
22
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| 1, 0, 2, -2, 0, 4, 0, -12, 0, 8, 12, 0, -48, 0, 16, 0, 120, 0, -160, 0, 32, -120, 0, 720, 0, -480, 0, 64, 0, -1680, 0, 3360, 0, -1344, 0, 128, 1680, 0, -13440, 0, 13440, 0, -3584, 0, 256, 0, 30240, 0, -80640, 0, 48384, 0, -9216, 0, 512, -30240, 0, 302400, 0, -403200, 0, 161280, 0, -23040, 0, 1024
(list; table; graph; listen)
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OFFSET
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0,3
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COMMENT
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Exponential Riordan array [exp(-x^2),2x]. [From Paul Barry (pbarry(AT)wit.ie), Jan 22 2009]
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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.
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LINKS
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T. D. Noe, 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].
R. D. Poshusta, Hermite Polynomials
Index entries for sequences related to Hermite polynomials
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FORMULA
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T(n, k)= ((-1)^((n-k)/2))*(2^k)*n!/(k!*((n-k)/2)!) if n-k is even and >=0, else 0.
E.g.f.: exp(-y^2+2*y*x).
T(n, k)=n!/(k!*2^((n-k)/2)((n-k)/2)!)2^((n+k)/2)cos(pi*(n-k)/2)(1+(-1)^(n+k))/2; T(n, k)=A001498((n+k)/2, (n-k)/2)*cos(pi*(n-k)/2)2^((n+k)/2)(1+(-1)^(n+k))/2; - Paul Barry (pbarry(AT)wit.ie), Aug 28 2005
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EXAMPLE
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[1], [0, 2], [ -2, 0, 4], [0, -12, 0, 8], [12, 0, -48, 0, 16], [0, 120, 0, -160, 0, 32], ... . Thus H_0(x)=1, H_1(x)=2*x, H_2(x)=-2+4*x^2, H_3(x)=-12*x+8*x^3, H_4(x)=12-48*x^2+16*x^4,...
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MAPLE
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with(orthopoly):for n from 0 to 10 do H(n, x):od;
T := proc(n, m) if n-m >= 0 and n-m mod 2 = 0 then ((-1)^((n-m)/2))*(2^m)*n!/(m!*((n-m)/2)!) else 0 fi; end;
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CROSSREFS
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Cf. A001814, A001816, A000321.
Without initial zeros, same as A059343.
Sequence in context: A138090 A138093 A138094 this_sequence A005881 A144458 A098268
Adjacent sequences: A060818 A060819 A060820 this_sequence A060822 A060823 A060824
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KEYWORD
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sign,tabl,nice
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AUTHOR
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Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 30 2001
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