A354621 - OEIS

%I #23 Mar 13 2026 19:06:27

%S 1,2,5,19,85,586,3583,28568,195449,1666786,18757980,161386953,

%T 1897428757,20910643255,186584844271,1896239913403,23753305611756,

%U 322385257985845,3291722491175736,43011227141438328,517673545204963277,5056620552149902641,65366993167319822971

%N Number of n-tuples (p_1, p_2, ..., p_n) of positive integers such that p_{i-1} <= p_i <= prime(i).

%C The number of n-tuples of primes with p_{i-1} <= p_i <= prime(i) give A000108.

%H Alois P. Heinz, <a href="/A354621/b354621.txt">Table of n, a(n) for n = 0..754</a>

%F a(n) = Sum_{j=0..n-1} a(j)*(-1)^(n+1-j)*binomial(prime(j+1),n-j) with a(0) = 1.

%F Sum_{n>=0} a(n)*x^n * (1-x)^prime(n+1) = 1.

%e a(0) = 1: ( ).

%e a(1) = 2: (1), (2).

%e a(2) = 5: (1,1), (1,2), (1,3), (2,2), (2,3).

%p b:= proc(n, i) option remember; `if`(n=0, 1,

%p       add(b(n-1, j), j=1..min(i, ithprime(n))))

%p     end:

%p a:= n-> b(n, infinity):

%p seq(a(n), n=0..23);

%p # Alternative:

%p a:= proc(n) option remember; `if`(n=0, 1, -add(a(j)*

%p       (-1)^(n-j)*binomial(ithprime(j+1), n-j), j=0..n-1))

%p     end:

%p seq(a(n), n=0..23);

%t a[n_] := a[n] = If[n == 0, 1, -Sum[a[j]*(-1)^(n - j)* Binomial[Prime[j + 1], n - j], {j, 0, n - 1}]];

%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Dec 28 2022, after second Maple program *)

%Y Cf. A000040, A000108, A325057.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Jul 08 2022