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%I #14 Jan 17 2019 04:05:59
%S 32,256,128,2144,256,4096,1024,512,16384,16384,85088,57088,299008,
%T 32768,524288,1172224,809344,21856256,950272,8388608,2097152,245600,
%U 865792,10231808,557056,15204352,4194304,134217728,21696512,15546368,1305935872,795410432,134217728
%N T(n,k) = numerator 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 (T(n,k)/A159983(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 | 32
%e 2 | 256 128
%e 3 | 2144 256 4096
%e 4 | 1024 512 16384 16384
%e 5 | 85088 57088 299008 32768 524288
%e 6 | 1172224 809344 21856256 950272 8388608 2097152
%e 7 | 245600 865792 10231808 557056 15204352 4194304 134217728
%e ...
%t T[n_, k_] := Numerator[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) := num(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. A159983 (denominators).
%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