A333371 Exponential convolution of primorial numbers (A002110) with themselves.

1, 4, 20, 132, 1116, 12420, 171300, 2884980, 56674380, 1289511300, 34769949060, 1063909626780, 37255008811020, 1470406699982220, 63114539746598340, 2936218980067393020, 150241360192861037100, 8497891914008911514100, 514514062115556069627060

Offset: 0

Author

Ilya Gutkovskiy, Mar 17 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..350
• Eric Weisstein's World of Mathematics, Primorial
• Index entries for sequences related to primorial numbers

Crossrefs

Cf. A002110, A014345, A052517, A062119, A136104, A333370.

Programs

Maple

p:= proc(n) option remember; `if`(n<1, 1, ithprime(n)*p(n-1)) end:
a:= n-> add(p(i)*p(n-i)*binomial(n, i), i=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 17 2020

Mathematica

primorial[n_] := Product[Prime[k], {k, 1, n}]; a[n_] := Sum[Binomial[n, k] primorial[k] primorial[n - k], {k, 0, n}]; Table[a[n], {n, 0, 18}]

Formula

E.g.f.: (Sum_{k>=0} prime(k)# * x^k / k!)^2, where prime()# = A002110.

a(n) = Sum_{k=0..n} binomial(n,k) * prime(k)# * prime(n-k)#.