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A058695 Number of ways to partition 2n+1 into positive integers.

%I A058695 #46 Feb 16 2025 08:32:43
%S A058695 1,3,7,15,30,56,101,176,297,490,792,1255,1958,3010,4565,6842,10143,
%T A058695 14883,21637,31185,44583,63261,89134,124754,173525,239943,329931,
%U A058695 451276,614154,831820,1121505,1505499,2012558,2679689,3554345,4697205,6185689,8118264,10619863
%N A058695 Number of ways to partition 2n+1 into positive integers.
%C A058695 A bisection of A000041, the other one is A058696.
%C A058695 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). - _Michael Somos_, Feb 16 2014
%C A058695 a(n) is the number of partitions of 3n-1 having n as a part, for n >=1.  Also, a(n+1) is the number of partitions of 3n having n as a part, for n >= 1. - _Clark Kimberling_, Mar 02 2014
%H A058695 Seiichi Manyama, <a href="/A058695/b058695.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..2000 from Robert Israel)
%H A058695 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A058695 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A058695 a(n) = A000041(2*n + 1).
%F A058695 Euler transform of period 16 sequence [ 3, 1, 2, 2, 2, 2, 3, 1, 3, 2, 2, 2, 2, 1, 3, 1, ...]. - _Michael Somos_, Apr 25 2003
%F A058695 G.f.: (Sum_{k>=0} x^A074377(k)) / (Product_{k>0} (1 - x^k))^2. - _Michael Somos_, Apr 25 2003
%F A058695 Expansion of f(x^1, x^7) / f(-x)^2 in powers of x where f() is a Ramanujan theta function. - _Michael Somos_, Feb 16 2014
%F A058695 Convolution of A000041 and A078408. - _Michael Somos_, Feb 16 2014
%e A058695 G.f. = 1 + 3*x + 7*x^2 + 15*x^3 + 30*x^4 + 56*x^5 + 101*x^6 + 176*x^7 + 297*x^8 + ...
%e A058695 G.f. = q^23 + 3*q^71 + 7*q^119 + 15*q^167 + 30*q^215 + 56*q^263 + 101*q^311 + ...
%p A058695 a:= n-> combinat[numbpart](2*n+1):
%p A058695 seq(a(n), n=0..42);  # _Alois P. Heinz_, Jan 29 2020
%t A058695 nn=100;Table[CoefficientList[Series[Product[1/(1-x^i),{i,1,nn}],{x,0,nn}],x][[2i]],{i,1,nn/2}] (* _Geoffrey Critzer_, Sep 28 2013 *)
%t A058695 (* also *)
%t A058695 Table[PartitionsP[2 n + 1], {n, 0, 40}] (* _Clark Kimberling_, Mar 02 2014 *)
%t A058695 (* also *)
%t A058695 Table[Count[IntegerPartitions[3 n - 1], p_ /; MemberQ[p, n]], {n, 20}]   (* _Clark Kimberling_, Mar 02 2014 *)
%o A058695 (PARI) {a(n) = if( n<0, 0, polcoeff( 1 / eta(x + O(x^(2*n + 2))), 2*n + 1))}; /* _Michael Somos_, Apr 25 2003 */
%o A058695 (PARI) a(n) = numbpart(2*n+1); \\ _Michel Marcus_, Sep 28 2013
%Y A058695 Cf. A000041, A058696, A074377, A078408.
%K A058695 nonn
%O A058695 0,2
%A A058695 _N. J. A. Sloane_, Dec 31 2000