This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A305871 #13 Feb 18 2022 07:55:36 %S A305871 2,2,1,2,2,-2,2,-2,4,-1,4,-7,10,-19,20,-20,34,-42,64,-100,126,-178, %T A305871 258,-326,464,-675,936,-1371,1888,-2550,3690,-5208,7292,-10467,14742, %U A305871 -20808,29610,-41586,59052,-84438,119602,-170153,242256,-343534,489550,-697815 %N A305871 -1 + Product_{n>=1} (1 + x^n)^a(n) = g.f. of A000040 (prime numbers). %C A305871 Inverse weigh transform of A000040. %H A305871 Alois P. Heinz, <a href="/A305871/b305871.txt">Table of n, a(n) for n = 1..3000</a> %H A305871 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a> %F A305871 Product_{n>=1} (1 + x^n)^a(n) = 1 + Sum_{k>=1} prime(k)*x^k. %e A305871 (1 + x)^2 * (1 + x^2)^2 * (1 + x^3) * (1 + x^4)^2 * (1 + x^5)^2 * (1 + x^6)^(-2) * ... * (1 + x^n)^a(n) * ... = 1 + 2*x + 3*x^2 + 5*x^3 + 7*x^4 + 11*x^5 + 13*x^6 + ... + A000040(k)*x^k + ... %p A305871 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, %p A305871 add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i))) %p A305871 end: %p A305871 a:= proc(n) option remember; ithprime(n)-b(n, n-1) end: %p A305871 seq(a(n), n=1..50); # _Alois P. Heinz_, Jun 13 2018 %t A305871 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, %t A305871 Sum[Binomial[a[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]]; %t A305871 a[n_] := a[n] = Prime[n] - b[n, n - 1]; %t A305871 Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Feb 18 2022, after _Alois P. Heinz_ *) %Y A305871 Cf. A000040, A030010, A030011, A061152. %K A305871 sign %O A305871 1,1 %A A305871 _Ilya Gutkovskiy_, Jun 12 2018