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A354621
Number of n-tuples (p_1, p_2, ..., p_n) of positive integers such that p_{i-1} <= p_i <= prime(i).
1
1, 2, 5, 19, 85, 586, 3583, 28568, 195449, 1666786, 18757980, 161386953, 1897428757, 20910643255, 186584844271, 1896239913403, 23753305611756, 322385257985845, 3291722491175736, 43011227141438328, 517673545204963277, 5056620552149902641, 65366993167319822971
OFFSET
0,2
COMMENTS
The number of n-tuples of primes with p_{i-1} <= p_i <= prime(i) give A000108.
LINKS
FORMULA
a(n) = Sum_{j=0..n-1} a(j)*(-1)^(n+1-j)*binomial(prime(j+1),n-j) with a(0) = 1.
Sum_{n>=0} a(n)*x^n * (1-x)^prime(n+1) = 1.
EXAMPLE
a(0) = 1: ( ).
a(1) = 2: (1), (2).
a(2) = 5: (1,1), (1,2), (1,3), (2,2), (2,3).
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1,
add(b(n-1, j), j=1..min(i, ithprime(n))))
end:
a:= n-> b(n, infinity):
seq(a(n), n=0..23);
# Alternative:
a:= proc(n) option remember; `if`(n=0, 1, -add(a(j)*
(-1)^(n-j)*binomial(ithprime(j+1), n-j), j=0..n-1))
end:
seq(a(n), n=0..23);
MATHEMATICA
a[n_] := a[n] = If[n == 0, 1, -Sum[a[j]*(-1)^(n - j)* Binomial[Prime[j + 1], n - j], {j, 0, n - 1}]];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 28 2022, after second Maple program *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 08 2022
STATUS
approved