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%I #11 Aug 02 2019 00:02:04
%S 1,2,9,58,475,4666,53116,684762,9833391,155341258,2673209561,
%T 49717424868,992847765988,21172798741316,479921234767976,
%U 11516219861132586,291523666535143823,7761036379846481206,216699016885046232187,6330257697841339549706,193043926318644060255531
%N a(n) = Sum_{k=0..n} (-1)^(n-k) * C(n,k) * Stirling1(n+1, k+1).
%H Robert Israel, <a href="/A247329/b247329.txt">Table of n, a(n) for n = 0..435</a>
%e Illustration of initial terms:
%e a(0) = 1*1 = 1 ;
%e a(1) = 1*1 + 1*1 = 2 ;
%e a(2) = 1*2 + 2*3 + 1*1 = 9 ;
%e a(3) = 1*6 + 3*11 + 3*6 + 1*1 = 58 ;
%e a(4) = 1*24 + 4*50 + 6*35 + 4*10 + 1*1 = 475 ;
%e a(5) = 1*120 + 5*274 + 10*225 + 10*85 + 5*15 + 1*1 = 4666 ;
%e a(6) = 1*720 + 6*1764 + 15*1624 + 20*735 + 15*175 + 6*21 + 1*1 = 53116 ;
%e a(7) = 1*5040 + 7*13068 + 21*13132 + 35*6769 + 35*1960 + 21*322 + 7*28 + 1*1 = 684762 ; ...
%p f:= proc(n) local k; add((-1)^(n-k)*binomial(n,k)*Stirling1(n+1,k+1),k=0..n); end proc:
%p map(f, [$0..30]); # _Robert Israel_, Aug 01 2019
%t Table[Sum[(-1)^(n-k) * Binomial[n,k] * StirlingS1[n+1, k+1],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Sep 29 2014 *)
%o (PARI) {Stirling1(n, k)=if(n==0, 1, n!*polcoeff(binomial(x, n), k))}
%o {a(n)=sum(k=0, n, (-1)^(n-k)*binomial(n,k)*Stirling1(n+1, k+1))}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A008275 (Stirling1 numbers), A211210.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 26 2014
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