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Exponential convolution of primorial numbers (A002110) with themselves.
2

%I #8 Mar 27 2020 19:57:11

%S 1,4,20,132,1116,12420,171300,2884980,56674380,1289511300,34769949060,

%T 1063909626780,37255008811020,1470406699982220,63114539746598340,

%U 2936218980067393020,150241360192861037100,8497891914008911514100,514514062115556069627060

%N Exponential convolution of primorial numbers (A002110) with themselves.

%H Alois P. Heinz, <a href="/A333371/b333371.txt">Table of n, a(n) for n = 0..350</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Primorial.html">Primorial</a>

%H <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>

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

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

%p p:= proc(n) option remember; `if`(n<1, 1, ithprime(n)*p(n-1)) end:

%p a:= n-> add(p(i)*p(n-i)*binomial(n, i), i=0..n):

%p seq(a(n), n=0..20); # _Alois P. Heinz_, Mar 17 2020

%t 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}]

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

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Mar 17 2020