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0, 1, 2, 7, 28, 139, 822, 5677, 44888, 400021, 3966970, 43328131, 516782292, 6682867087, 93130824878, 1391321096089, 22181459914672, 375880800693097, 6746469047955378, 127851581333528191, 2551039715319388940, 53457519928692619411, 1173770856436282074982, 26948387795024752862917, 645694707721735535710728, 16117771962578155161812989
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = Sum_{k = 0..floor((n-1)/2)} (n-2*k)!*binomial(n-k-1,k).
O.g.f.: Sum_{n >= 1} n!*x^n/(1 - x^2)^n = x + 2*x^2 + 7*x^3 + 28*x^4 + ....
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MAPLE
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A276080 := proc (n) add((n-2*k)*factorial(n-k-1)/factorial(k), k = 0..floor((1/2)*n-1/2)) end proc:
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MATHEMATICA
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Map[If[# == 1, 0, Total[FactorInteger[#] /. {p_, e_} /; p > 1 :> e PrimePi[p]!]] &, Nest[Append[#, (Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ #[[-1]]) #[[-2]]] &, {1, 2}, 24]] (* Michael De Vlieger, Dec 24 2017 *)
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PROG
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(Scheme)
;; A more practical standalone program, that uses memoization-macro definec:
(define (A276080 n) (sum_factorials_times_elements_in (A206296as_index_lists n)))
(definec (A206296as_index_lists n) (cond ((zero? n) (list)) ((= 1 n) (list 1)) (else (map + (cons 0 (A206296as_index_lists (- n 1))) (append (A206296as_index_lists (- n 2)) (list 0 0))))))
(define (sum_factorials_times_elements_in nums) (let loop ((s 0) (nums nums) (i 2) (f 1)) (cond ((null? nums) s) (else (loop (+ s (* (car nums) f)) (cdr nums) (+ 1 i) (* i f))))))
(Python)
from sympy import factorint, factorial as f, prime, primepi
from operator import mul
from functools import reduce
def a003961(n):
F=factorint(n)
return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**F[i] for i in F])
def a276075(n):
F=factorint(n)
return 0 if n==1 else sum([F[i]*f(primepi(i)) for i in F])
l=[1, 2]
L=[0, 1]
for n in range(2, 11):
l.append(a003961(l[n - 1])*l[n - 2])
L.append(a276075(l[n]))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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