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A065048 Largest unsigned Stirling number of the first kind: max_k(s(n+1,k)); i.e., largest coefficient of polynomial x*(x+1)*(x+2)*(x+3)*...*(x+n).

%I A065048 #39 Jun 11 2025 07:34:31
%S A065048 1,1,3,11,50,274,1764,13132,118124,1172700,12753576,150917976,
%T A065048 1931559552,26596717056,392156797824,6165817614720,102992244837120,
%U A065048 1821602444624640,34012249593822720,668609730341153280,13803759753640704000,298631902863216384000
%N A065048 Largest unsigned Stirling number of the first kind: max_k(s(n+1,k)); i.e., largest coefficient of polynomial x*(x+1)*(x+2)*(x+3)*...*(x+n).
%C A065048 n! <= a(n) <= (n+1)!; n <= a(n+1)/a(n) <= (n+1). - _Max Alekseyev_, Jul 17 2019
%H A065048 Robert Israel, <a href="/A065048/b065048.txt">Table of n, a(n) for n = 0..448</a>
%F A065048 For n in the interval [A309237(k)-1, A309237(k+1)-2], a(n) = |Stirling1(n+1,k)|. - _Max Alekseyev_, Jul 17 2019
%e A065048 a(4)=50 since polynomial is x^4 + 10*x^3 + 35*x^2 + 50*x + 24.
%p A065048 P:= x: A[0]:= 1:
%p A065048 for n from 1 to 50 do
%p A065048   P:= expand(P*(x+n));
%p A065048   A[n]:= max(coeffs(P,x));
%p A065048 od:
%p A065048 seq(A[i],i=0..50); # _Robert Israel_, Jul 04 2016
%t A065048 a[n_] := Max[Array[Abs[StirlingS1[n+1, #]]&, n+1]];
%t A065048 Array[a, 100, 0] (* _Griffin N. Macris_, Jul 03 2016 *)
%o A065048 (PARI) a(n) = if (n==0, 1, vecmax(vector(n, k, abs(stirling(n+1, k, 1))))); \\ _Michel Marcus_, Jul 04 2016; corrected Jun 12 2022
%o A065048 (Python)
%o A065048 from collections import Counter
%o A065048 def A065048(n):
%o A065048     c = {1:1}
%o A065048     for k in range(1,n+1):
%o A065048         d = Counter()
%o A065048         for j in c:
%o A065048             d[j] += k*c[j]
%o A065048             d[j+1] += c[j]
%o A065048         c = d
%o A065048     return max(c.values()) # _Chai Wah Wu_, Jan 31 2024
%Y A065048 Cf. A000254, A000399, A002870, A008275, A309237.
%K A065048 nonn
%O A065048 0,3
%A A065048 _Henry Bottomley_, Nov 06 2001