A023626 Self-convolution of (1, p(1), p(2), ...).

1, 4, 10, 22, 43, 80, 137, 222, 343, 508, 737, 1030, 1411, 1888, 2477, 3198, 4059, 5096, 6297, 7702, 9327, 11176, 13301, 15682, 18355, 21344, 24673, 28358, 32411, 36896, 41769, 47082, 52883, 59148, 65937, 73298, 81251, 89776, 98957

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

p(1),p(2),p(3)... are the prime numbers (A000040). The analogous sequence for the partition numbers is A048574.

Examples

			G.f. = x + 4*x^2 + 10*x^3 + 22*x^4 + 43*x^5 + 80*x^6 + 137*x^7 + ...
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A008578, A014342, A048574.

Programs

Haskell

a023626 n = a023626_list !! (n-2)
a023626_list = f a000040_list [1] where
   f (p:ps) rs = (sum $ zipWith (*) rs a008578_list) : f ps (p : rs)
-- Reinhard Zumkeller, Nov 09 2015

Mathematica

z = 100; p = Join[{1}, Prime[Range[z]]];
a[n_] := Sum[p[[i]] p[[n - i + 1]], {i, 1, n}];
Table[a[n], {n, 1, z}]  (* Clark Kimberling, Dec 01 2016 *)
a[ n_] := If[ n < 1, 0, SeriesCoefficient[ (1 + O[x]^n + Sum[ Prime[k] x^k, {k, n - 1}])^2, {x, 0, n - 1}]]; (* Michael Somos, Dec 01 2016 *)
Table[With[{c=Join[{1},Prime[Range[n]]]},ListConvolve[c,c]],{n,0,40}]// Flatten (* Harvey P. Dale, Oct 19 2018 *)

Formula

G.f: x*(1+b(x))^2 = (c(x)^2)/x, where b(x) and c(x) are respectively the g.f. of A000040 and A008578. - Mario C. Enriquez, Dec 10 2016