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Stirling numbers of the first kind: Difference between revisions
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19710908747055261109287881673376044669240511161402863823515728791076863288440277983854056472903481625299174865860036734731122707870406148096000000000000000000
(158 digits, k = 5)</pre>
=={{header|Nim}}==
A simple implementation using the recursive definition is enough to complete the task.
<lang Nim>import sequtils, strutils
proc s1(n, k: Natural): Natural =
if k == 0: return ord(n == 0)
if k > n: return 0
s1(n - 1, k - 1) + (n - 1) * s1(n - 1, k)
echo " k ", toSeq(0..12).mapIt(($it).align(2)).join(" ")
echo " n"
for n in 0..12:
stdout.write ($n).align(2)
for k in 0..n:
stdout.write ($s1(n, k)).align(10)
stdout.write '\n'</lang>
{{out}}
<pre>k 0 1 2 3 4 5 6 7 8 9 10 11 12
n
0 1
1 0 1
2 0 1 1
3 0 2 3 1
4 0 6 11 6 1
5 0 24 50 35 10 1
6 0 120 274 225 85 15 1
7 0 720 1764 1624 735 175 21 1
8 0 5040 13068 13132 6769 1960 322 28 1
9 0 40320 109584 118124 67284 22449 4536 546 36 1
10 0 362880 1026576 1172700 723680 269325 63273 9450 870 45 1
11 0 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1
12 0 39916800 120543840 150917976 105258076 45995730 13339535 2637558 357423 32670 1925 66 1</pre>
To find the maximum value of S1(n, k) for n = 100, we need to use a third party library such as “bignum”. But this is not enough. To avoid multiple computation of the same terms, we have to use a cache. Note also that this method has its limits as it depends on the allowed depth of recursion. Here, this is not a problem and the result is obtained almost instantaneously.
{{libheader|bignum}}
<lang Nim>import tables
import bignum
var cache: Table[(Natural, Natural), Int]
proc s1(n, k: Natural): Int =
if k == 0: return newInt(ord(n == 0))
if k > n: return newInt(0)
if (n, k) in cache: return cache[(n, k)]
result = s1(n - 1, k - 1) + (n - 1) * s1(n - 1, k)
cache[(n, k)] = result
var max = newInt(-1)
for k in 0..100:
let s = s1(100, k)
if s > max: max = s
else: break
echo "Maximum Stirling number of the first kind with n = 100:"
echo max</lang>
{{out}}
<pre>Maximum Stirling number of the first kind with n = 100:
19710908747055261109287881673376044669240511161402863823515728791076863288440277983854056472903481625299174865860036734731122707870406148096000000000000000000</pre>
=={{header|Perl}}==
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