A276080 a(n) = A276075(A206296(n)).

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

Offset: 0

Author

Antti Karttunen, Aug 18 2016

Keywords

• nonn

Crossrefs

Cf. A000045, A000142, A206296, A276075.

Cf. also A276081, A001338, A007188.

Programs

Maple

A276080 := proc (n) add((n-2*k)*factorial(n-k-1)/factorial(k), k = 0..floor((1/2)*n-1/2)) end proc:
seq(A276080(n), n = 0..25); # Peter Bala, Dec 24 2017

Mathematica

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 *)

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]))
print(L) # Indranil Ghosh, Jun 21 2017

Scheme

(define (A276080 n) (A276075 (A206296 n)))
;; 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))))))

Formula

a(n) = A276075(A206296(n)).

From Peter Bala, Dec 24 2017: (Start)

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 + ....

Cf. A001339(n) = A276075(A007188(n)) for n >= 1, with o.g.f. Sum_{n >= 0} n!*x^n/(1 - x)^n. (End)