These are the asymptotics of the sum of the powers of Pascal's

                     triangles for powers between 2 to , 8

                               The asymtotics of

                               n
                             -----
                              \                  2
                               )   binomial(n, k)
                              /
                             -----
                             k = 0

                          up to order, 5, is equal to

                                n      1/2
0.5641895835477562869480794517 4  (1/n)

    /     1      1         5         21         399   \
    |1 - --- + ------ + ------- - -------- - ---------|
    |    8 n        2         3          4           5|
    \          128 n    1024 n    32768 n    262144 n /

                               The asymtotics of

                               n
                             -----
                              \                  3
                               )   binomial(n, k)
                              /
                             -----
                             k = 0

                          up to order, 5, is equal to

                               n /     1      1       1       1        11   \
0.367552596947861366340884333 8  |1 - --- + ----- + ----- + ------ + -------|/n
                                 |    3 n       2       3        4         5|
                                 \          27 n    81 n    243 n    2187 n /

                               The asymtotics of

                               n
                             -----
                              \                  4
                               )   binomial(n, k)
                              /
                             -----
                             k = 0

                          up to order, 5, is equal to

                               n      (3/2)
0.25397454373696387914305321 16  (1/n)

    /     9       65       205       4779        48135   \
    |1 - ---- + ------ + ------- + --------- - ----------|
    |    16 n        2         3           4            5|
    \           512 n    8192 n    524288 n    8388608 n /

                               The asymtotics of

                               n
                             -----
                              \                  5
                               )   binomial(n, k)
                              /
                             -----
                             k = 0

                          up to order, 5, is equal to

                               n /     4      7       2         3        112  \
0.18124884334800559660450082 32  |1 - --- + ----- + ------ - ------- - -------|
                                 |    5 n       2        3         4         5|
                                 \          25 n    125 n    3125 n    3125 n /

       /  2
      /  n
     /

                               The asymtotics of

                               n
                             -----
                              \                  6
                               )   binomial(n, k)
                              /
                             -----
                             k = 0

                          up to order, 5, is equal to

                               n      (5/2)
0.13201542623099540327200772 64  (1/n)

    /     25     1715       2905       157087       38920847  \
    |1 - ---- + ------- - -------- - ---------- - ------------|
    |    24 n         2          3            4              5|
    \           3456 n    82944 n    7962624 n    573308928 n /

                               The asymtotics of

                               n
                             -----
                              \                  7
                               )   binomial(n, k)
                              /
                             -----
                             k = 0

                          up to order, 5, is equal to

                                 n
0.097519473726265526957529725 128

    /     9     38       50       81        1543  \   /  3
    |1 - --- + ----- - ------ - ------- - --------|  /  n
    |    7 n       2        3         4          5| /
    \          49 n    343 n    2401 n    16807 n /

                               The asymtotics of

                               n
                             -----
                              \                  8
                               )   binomial(n, k)
                              /
                             -----
                             k = 0

                          up to order, 5, is equal to

                                 n      (7/2)
0.072783919122571767633118288 256  (1/n)

    /     49     2289      21819       210357       28109823  \
    |1 - ---- + ------- - -------- - ---------- - ------------|
    |    32 n         2          3            4              5|
    \           2048 n    65536 n    8388608 n    268435456 n /

             Everything is rigorous, but the constants in front are

                  non-rigorous (yet fairly reliable) estimates

                    This took, 139.722, seconds of CPU time