A121572 Subprimorials: inverse binomial transform of primorials (A002110).

1, 1, 3, 17, 119, 1509, 18799, 342397, 6340263, 151918421, 4619754311, 140219120601, 5396354613583, 221721908976697, 9431597787000999, 447473598316521449, 24163152239530299719, 1444153946379288324477, 87200644323074509092943, 5929294512595059362045041

Offset: 0

Author

Franklin T. Adams-Watters, Aug 08 2006

Keywords

• nonn

Comments

By analogy with subfactorials, which are the inverse binomial transform of the factorials.

Examples

			a(3) = 30 - 3*6 + 3*2 - 1 = 17.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..350

Crossrefs

Cf. A002110, A000166, A136104.

See A079266 for a different definition of subprimorial.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, ithprime(n)*b(n-1)) end:
a:= n-> add(binomial(n,k)*b(k)*(-1)^(n-k), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 19 2016

Mathematica

b[n_] := b[n] = If[n==0, 1, Prime[n]*b[n-1]]; a[n_] := Sum[Binomial[n, k]* b[k]*(-1)^(n-k), {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 13 2017, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) C(n,k) Prime(k)#, where p# is p primorial and Prime(0)# = 1.

A007318^(-1) * A002110. - Gary W. Adamson, Dec 14 2007

Extensions

More terms from R. J. Mathar, Sep 18 2007

Edited by N. J. A. Sloane, May 15 2008 at the suggestion of R. J. Mathar