A159983 - OEIS

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).

%I #14 Jan 17 2019 04:06:19

%S 3,15,15,105,21,525,45,35,1575,2205,3465,3465,24255,3465,72765,45045,

%T 45045,1576575,85995,945945,297297,9009,45045,675675,45045,1486485,

%U 495495,19324305,765765,765765,80405325,58963905,11792781,1738165,65702637,78217425

%N 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).

%C 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.

%H Robert Delbourgo and David Elliott, <a href="https://doi.org/10.1063/1.3141534">Inverse momentum expectation values for hydrogenic systems</a>, J. Math. Phys. 50, 062107 (2009); <a href="https://arxiv.org/abs/0904.4288">arXiv:0904.4288 [math-ph]</a>, 2009.

%e Triangle begins:

%e n\k | 0 1 2 3 4 5 6

%e -------------------------------------------------------

%e 1 | 3

%e 2 | 15 15

%e 3 | 105 21 525

%e 4 | 45 35 1575 2205

%e 5 | 3465 3465 24255 3465 72765

%e 6 | 45045 45045 1576575 85995 945945 297297

%e 7 | 9009 45045 675675 45045 1486485 495495 19324305

%e ...

%t 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}]]

%t Table[T[n, k], {n, 1, 20}, {k, 0, n - 1}] // Flatten

%t (* _Franck Maminirina Ramaharo_, Jan 16 2019 *)

%o (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))$

%o create_list(T(n, k), n, 1, 20, k, 0, n - 1);

%o /* _Franck Maminirina Ramaharo_, Jan 16 2019 */

%Y Cf. A003671, A003676.

%Y Cf. A159982 (numerators).

%K frac,nonn,tabl,easy

%O 1,1

%A _Jonathan Vos Post_, Apr 28 2009

%E Edited and extended by _Franck Maminirina Ramaharo_, Jan 16 2019