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A177453
Partial sums of A001863.
1
0, 1, 5, 31, 267, 3027, 42599, 715191, 13942995, 309522515, 7707841015, 212783127799, 6449579387715, 212939326904131, 7606688596589431, 292321288041079671, 12025358303201356019, 527265684696785414387
OFFSET
1,3
COMMENTS
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).
FORMULA
a(n) = Sum_{i=1..n} A001863(i).
EXAMPLE
a(5) = 0 + 1 + 4 + 26 + 236 = 267 = 3 * 89.
MAPLE
A001863 := proc(n) if n = 1 then 0; else add( (n-2)!*n^k/k!, k=0..n-2) ; end if; end proc:
A177453 := proc(n) add(A001863(i), i=0..n) ; end proc: seq(A177453(n), n=1..20) ; # R. J. Mathar, May 28 2010
MATHEMATICA
Accumulate[Table[Sum[(n-2)! n^k/k!, {k, 0, n-2}], {n, 20}]] (* Harvey P. Dale, Jun 19 2016 *)
PROG
(Python)
from math import comb
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Vos Post, May 09 2010
EXTENSIONS
Extended by R. J. Mathar, May 28 2010
STATUS
approved