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

1,2

Comments

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

Links

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

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A000040, A008578, A014342, A048574.

Sequence in context: A008256 A006001 A034357 * A048574 A052837 A052821

Adjacent sequences: A023623 A023624 A023625 * A023627 A023628 A023629

Keyword

nonn

Author

Clark Kimberling

Status

approved