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A159983
T(n,k) = denominator of 2*Pi*Sum_{j=0..n-k-1} ((-1)^j*n*(k + j + 2)*(n + k +j)!*(k + j)!^2)/((n - k - j - 1)!*(2*k + j + 1)!*j!*Gamma(k + j + 3/2)*Gamma(k + j + 5/2)), triangle read by rows (n >= 1, 0 <= k <= n - 1).
1
3, 15, 15, 105, 21, 525, 45, 35, 1575, 2205, 3465, 3465, 24255, 3465, 72765, 45045, 45045, 1576575, 85995, 945945, 297297, 9009, 45045, 675675, 45045, 1486485, 495495, 19324305, 765765, 765765, 80405325, 58963905, 11792781, 1738165, 65702637, 78217425
OFFSET
1,1
COMMENTS
Let a_0 and h denote Bohr radius A003671 and Planck constant A003676, respectively. Then (A159982(n,k)/T(n,k))*(n*a_0/h) is the expectation value of any inverse momentum function, where n and k are quantum numbers which are integers obeying n > k >= 0.
LINKS
Robert Delbourgo and David Elliott, Inverse momentum expectation values for hydrogenic systems, J. Math. Phys. 50, 062107 (2009); arXiv:0904.4288 [math-ph], 2009.
EXAMPLE
Triangle begins:
n\k | 0 1 2 3 4 5 6
-------------------------------------------------------
1 | 3
2 | 15 15
3 | 105 21 525
4 | 45 35 1575 2205
5 | 3465 3465 24255 3465 72765
6 | 45045 45045 1576575 85995 945945 297297
7 | 9009 45045 675675 45045 1486485 495495 19324305
...
MATHEMATICA
T[n_, k_] := Denominator[2*Pi*Sum[((-1)^j*n*(k + j + 2)*(n + k +j)!*(k + j)!^2)/((n - k - j - 1)!*(2*k + j + 1)!*j!*Gamma[k + j + 3/2]*Gamma[k + j + 5/2]), {j, 0, n - k - 1}]]
Table[T[n, k], {n, 1, 20}, {k, 0, n - 1}] // Flatten
(* Franck Maminirina Ramaharo, Jan 16 2019 *)
PROG
(Maxima) T(n, k) := denom(2*%pi*sum(((-1)^j*n*(k + j + 2)*(n + k + j)!*(k + j)!^2)/((n - k - j - 1)!*(2*k + j + 1)!*j!*gamma(k + j + 3/2)*gamma(k + j + 5/2)), j, 0, n - k - 1))$
create_list(T(n, k), n, 1, 20, k, 0, n - 1);
/* Franck Maminirina Ramaharo, Jan 16 2019 */
CROSSREFS
Cf. A159982 (numerators).
Sequence in context: A289403 A289459 A257115 * A133227 A237628 A074043
KEYWORD
frac,nonn,tabl,easy
AUTHOR
Jonathan Vos Post, Apr 28 2009
EXTENSIONS
Edited and extended by Franck Maminirina Ramaharo, Jan 16 2019
STATUS
approved