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).

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

Table of n, a(n) for n=1..36.

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. A003671, A003676.

Cf. A159982 (numerators).

Sequence in context: A289403 A289459 A257115 * A133227 A237628 A074043

Adjacent sequences: A159980 A159981 A159982 * A159984 A159985 A159986

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