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

1, 3, 7, 15, 30, 56, 101, 176, 297, 490, 792, 1255, 1958, 3010, 4565, 6842, 10143, 14883, 21637, 31185, 44583, 63261, 89134, 124754, 173525, 239943, 329931, 451276, 614154, 831820, 1121505, 1505499, 2012558, 2679689, 3554345, 4697205, 6185689, 8118264, 10619863

Offset: 0

Author

N. J. A. Sloane, Dec 31 2000

Keywords

• nonn

Comments

A bisection of A000041, the other one is A058696.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). - Michael Somos, Feb 16 2014

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

Examples

			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 + ...
G.f. = q^23 + 3*q^71 + 7*q^119 + 15*q^167 + 30*q^215 + 56*q^263 + 101*q^311 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Robert Israel)
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A000041, A058696, A074377, A078408.

Programs

Maple

a:= n-> combinat[numbpart](2*n+1):
seq(a(n), n=0..42);  # Alois P. Heinz, Jan 29 2020

Mathematica

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 *)
(* also *)
Table[PartitionsP[2 n + 1], {n, 0, 40}] (* Clark Kimberling, Mar 02 2014 *)
(* also *)
Table[Count[IntegerPartitions[3 n - 1], p_ /; MemberQ[p, n]], {n, 20}]   (* Clark Kimberling, Mar 02 2014 *)

PARI

{a(n) = if( n<0, 0, polcoeff( 1 / eta(x + O(x^(2*n + 2))), 2*n + 1))}; /* Michael Somos, Apr 25 2003 */

PARI

a(n) = numbpart(2*n+1); \\ Michel Marcus, Sep 28 2013

Formula

a(n) = A000041(2*n + 1).

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

G.f.: (Sum_{k>=0} x^A074377(k)) / (Product_{k>0} (1 - x^k))^2. - Michael Somos, Apr 25 2003

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

Convolution of A000041 and A078408. - Michael Somos, Feb 16 2014