A177453 - OEIS

%I #15 Apr 26 2023 10:08:17

%S 0,1,5,31,267,3027,42599,715191,13942995,309522515,7707841015,

%T 212783127799,6449579387715,212939326904131,7606688596589431,

%U 292321288041079671,12025358303201356019,527265684696785414387

%N Partial sums of A001863.

%C Partial sums of normalized total height of rooted trees with n nodes. The subsequence of primes in the partial sums begins: 5, 31, no more through a(15).

%F a(n) = Sum_{i=1..n} A001863(i).

%e a(5) = 0 + 1 + 4 + 26 + 236 = 267 = 3 * 89.

%p A001863 := proc(n) if n = 1 then 0; else add( (n-2)!*n^k/k!,k=0..n-2) ; end if; end proc:

%p A177453 := proc(n) add(A001863(i),i=0..n) ; end proc: seq(A177453(n),n=1..20) ; # _R. J. Mathar_, May 28 2010

%t Accumulate[Table[Sum[(n-2)! n^k/k!,{k,0,n-2}],{n,20}]] (* _Harvey P. Dale_, Jun 19 2016 *)

%o (Python)

%o from math import comb

%o def A177453(n): return sum(((sum(comb(i,k)*(i-k)**(i-k)*k**k for k in range(1,(i+1>>1)))<<1) + (0 if i&1 else comb(i,m:=i>>1)*m**i))//i//(i-1) for i in range(2,n+1)) # _Chai Wah Wu_, Apr 25-26 2023

%Y Cf. A000435, A001864, A001863.

%K nonn

%O 1,3

%A _Jonathan Vos Post_, May 09 2010

%E Extended by _R. J. Mathar_, May 28 2010