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A161077 Number of partitions of n into primes or 1 where every part appears at least 2 times.

%I A161077 #19 Oct 07 2024 00:43:11
%S A161077 0,1,1,2,1,4,2,5,5,8,6,12,10,16,16,22,21,31,30,40,42,53,55,71,73,90,
%T A161077 96,116,123,149,157,186,200,234,250,292,312,360,388,443,476,543,584,
%U A161077 660,712,800,862,967,1041,1160,1252,1390,1497,1659,1786,1970,2124,2336,2513,2760
%N A161077 Number of partitions of n into primes or 1 where every part appears at least 2 times.
%H A161077 R. H. Hardin, <a href="/A161077/b161077.txt">Table of n, a(n) for n = 1..1000</a>
%F A161077 G.f.: -1+(1+x^2/(1-x))*Product_{j>=1} ( 1+x^(2*p(j))/(1-x^(p(j))) ), where p(j) is the j-th prime. - _Emeric Deutsch_, Jun 27 2009
%e A161077 a(8)=5 because we have 3311, 2222, 22211, 221111, and 1^8. - _Emeric Deutsch_, Jun 27 2009
%p A161077 g := -1+(1+x^2/(1-x))*(product(1+x^(2*ithprime(j))/(1-x^ithprime(j)), j = 1 .. 20)): gser := series(g, x = 0, 70): seq(coeff(gser, x, n), n = 1 .. 60); # _Emeric Deutsch_, Jun 27 2009
%t A161077 nmax = 100; Rest[CoefficientList[Series[-1 + (1 + x^2/(1-x)) * Product[1 + x^(2*Prime[k]) / (1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Nov 28 2020 *)
%K A161077 nonn
%O A161077 1,4
%A A161077 _R. H. Hardin_, Jun 02 2009
%E A161077 Definition edited to "primes or 1" by _R. H. Hardin_, Jun 22 2009