|
Search: id:A067147
|
|
|
| A067147 |
|
Triangle of coefficients for expressing x^n in terms of Hermite polynomials. |
|
+0
6
|
|
| 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, 0, 332640, 0, 277200, 0
(list; table; graph; listen)
|
|
|
OFFSET
|
0,4
|
|
|
COMMENT
|
x^n = 1/2^n * Sum (a(n,k)*H_k(x)), k=0..n
|
|
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
|
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].
Index entries for sequences related to Hermite polynomials
|
|
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 DELEHAM (kolotoko(AT)wanadoo.fr), 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 DELEHAM (kolotoko(AT)wanadoo.fr), 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 (pbarry(AT)wit.ie), Aug 28 2005
Exponential Riordan array (e^(x^2),x). - Paul Barry (pbarry(AT)wit.ie), Sep 12 2006
|
|
EXAMPLE
|
1; 0,1; 2,0,1; 0,6,0,1; 12,0,12,0,1; ...
|
|
CROSSREFS
|
Row sums give A047974. Columns 0-2: A001813, A000407, A001814. Cf. A048854, A060821.
Sequence in context: A137526 A137525 A109187 this_sequence A112227 A136579 A076694
Adjacent sequences: A067144 A067145 A067146 this_sequence A067148 A067149 A067150
|
|
KEYWORD
|
nonn,tabl
|
|
AUTHOR
|
Christian G. Bower (bowerc(AT)usa.net), Jan 03 2002
|
|
|
Search completed in 0.002 seconds
|
