%I #30 May 08 2021 23:05:25

%S 0,1,2,7,28,139,822,5677,44888,400021,3966970,43328131,516782292,

%T 6682867087,93130824878,1391321096089,22181459914672,375880800693097,

%U 6746469047955378,127851581333528191,2551039715319388940,53457519928692619411,1173770856436282074982,26948387795024752862917,645694707721735535710728,16117771962578155161812989

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

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

%F From _Peter Bala_, Dec 24 2017: (Start)

%F a(n) = Sum_{k = 0..floor((n-1)/2)} (n-2*k)!*binomial(n-k-1,k).

%F O.g.f.: Sum_{n >= 1} n!*x^n/(1 - x^2)^n = x + 2*x^2 + 7*x^3 + 28*x^4 + ....

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

%p A276080 := proc (n) add((n-2*k)*factorial(n-k-1)/factorial(k), k = 0..floor((1/2)*n-1/2)) end proc:

%p seq(A276080(n), n = 0..25); # _Peter Bala_, Dec 24 2017

%t 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 *)

%o (Scheme)

%o (define (A276080 n) (A276075 (A206296 n)))

%o ;; A more practical standalone program, that uses memoization-macro definec:

%o (define (A276080 n) (sum_factorials_times_elements_in (A206296as_index_lists n)))

%o (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))))))

%o (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))))))

%o (Python)

%o from sympy import factorint, factorial as f, prime, primepi

%o from operator import mul

%o from functools import reduce

%o def a003961(n):

%o F=factorint(n)

%o return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**F[i] for i in F])

%o def a276075(n):

%o F=factorint(n)

%o return 0 if n==1 else sum([F[i]*f(primepi(i)) for i in F])

%o l=[1, 2]

%o L=[0, 1]

%o for n in range(2, 11):

%o l.append(a003961(l[n - 1])*l[n - 2])

%o L.append(a276075(l[n]))

%o print(L) # _Indranil Ghosh_, Jun 21 2017

%Y Cf. A000045, A000142, A206296, A276075.

%Y Cf. also A276081, A001338, A007188.

%K nonn

%O 0,3

%A _Antti Karttunen_, Aug 18 2016