A159982 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).

32, 256, 128, 2144, 256, 4096, 1024, 512, 16384, 16384, 85088, 57088, 299008, 32768, 524288, 1172224, 809344, 21856256, 950272, 8388608, 2097152, 245600, 865792, 10231808, 557056, 15204352, 4194304, 134217728, 21696512, 15546368, 1305935872, 795410432, 134217728

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..33.

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  |      32
   2  |     256    128
   3  |    2144    256     4096
   4  |    1024    512    16384  16384
   5  |   85088  57088   299008  32768   524288
   6  | 1172224 809344 21856256 950272  8388608 2097152
   7  |  245600 865792 10231808 557056 15204352 4194304 134217728
   ...

Mathematica

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}]]
Table[T[n, k], {n, 1, 20}, {k, 0, n - 1}] // Flatten
(* Franck Maminirina Ramaharo, Jan 16 2019 *)

Prog

(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))$
create_list(T(n, k), n, 1, 20, k, 0, n - 1);
/* Franck Maminirina Ramaharo, Jan 16 2019 */

Crossrefs

Cf. A003671, A003676.

Cf. A159983 (denominators).

Sequence in context: A300565 A231801 A250280 * A195592 A097243 A357788

Adjacent sequences: A159979 A159980 A159981 * A159983 A159984 A159985

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