A078408 Number of ways to partition 2n+1 into distinct positive integers.

1, 2, 3, 5, 8, 12, 18, 27, 38, 54, 76, 104, 142, 192, 256, 340, 448, 585, 760, 982, 1260, 1610, 2048, 2590, 3264, 4097, 5120, 6378, 7917, 9792, 12076, 14848, 18200, 22250, 27130, 32992, 40026, 48446, 58499, 70488, 84756, 101698, 121792, 145578, 173682

Offset: 0

Author

N. J. A. Sloane, Dec 27 2002

Keywords

• nonn
• easy

Comments

a(n) is also the number of partitions of 2n+1 in which all parts are odd, due to Euler's partition theorem. See A000009. - Wolfdieter Lang, Jul 08 2012

Examples

			a(3) = 5 because 7 = 1+6 = 2+5 = 3+4 = 1+2+4 (partitions into distinct parts) and 7 = 1+1+5 = 1+3+3 = 1+1+1+1+3 = 1+1+1+1+1+1+1 (partitions into odd parts). [_Wolfdieter Lang_, Jul 08 2012]
G.f. = 1 + 2*x + 3*x^2 + 5*x^3 + 8*x^4 + 12*x^5 + 18*x^6 + 27*x^7 + 38*x^8 + ...
G.f. = q^25 + 2*q^73 + 3*q^121 + 5*q^169 + 8*q^217 + 12*q^265 + 18*q^313 + ...
		

References

• G. E. Andrews, The Theory of Partitions, Cambridge University Press, 1998, p. 19.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Reinhard Zumkeller)
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A035294, A078409, A078410.

Cf. A005408, A000009.

Programs

Haskell

import Data.MemoCombinators (memo2, integral)
a078408 n = a078408_list !! n
a078408_list = f 1 where
   f x = (p' 1 x) : f (x + 2)
   p' = memo2 integral integral p
   p _ 0 = 1
   p k m = if m < k then 0 else p' k (m - k) + p' (k + 2) m
-- Reinhard Zumkeller, Nov 27 2015

Maple

G := 1/(1 - x)*add(x^floor(3*n/2)/mul(1 - x^k, k = 1..n), n = 0..50):
S := series(G, x, 76):
seq(coeff(S, x, j), j = 0..75); # Peter Bala, Feb 04 2021

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] / QPochhammer[ x], {x, 0, 2 n + 1}]; (* Michael Somos, Oct 06 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, n = 2*n + 1; A = x * O(x^n); polcoeff( eta(x^2 + A) / eta(x + A), n))};

Formula

a(n) = t(2*n+1, 0), t as defined in A079211.

Euler transform of period 16 sequence [ 2, 0, 1, 1, 1, 1, 2, 0, 2, 1, 1, 1, 1, 0, 2, 0, ...]. - Michael Somos, Mar 04 2003

a(n) = A000009(2*n+1). G.f. of A000009: 1/[(1 - x)*(1 - x^3)*(1 - x^5)*...] - Jon Perry, May 27 2004

Expansion of f(x, x^7) / f(-x, -x^2) where f(, ) is Ramanujan's general theta function. - Michael Somos, Oct 06 2015

From Peter Bala, Feb 04 2021: (Start)

G.f.: Sum_{n >= 0} x^n/Product_{k = 1..2*n+1} 1 - x^k. Replace q with q^2 and set t = q in Andrews, equation 2.2.5, p. 19, and then take the odd part of the series.

G.f.: 1/(1 - x)*Sum_{n >= 0} x^floor(3*n/2)/Product_{k = 1..n} (1 - x^k). (End)

a(n) = A282893(n+1) + A238478(n+1) = A035294(n+1) - A238479(n+1). - Mathew Englander, May 24 2023

G.f.: Product_{n >= 1} (1 - q^(8*n))*(1 + q^(8*n-1))*(1 + q^(8*n-7))/(1 - q^n). - Peter Bala, Dec 30 2024

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(11/4)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jul 06 2025

Extensions

More terms from Reinhard Zumkeller, Dec 28 2002