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

0,2

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

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

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(2n+1). G.f.: 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

Example

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 + ...

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 *)

Prog

(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))};
(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

Crossrefs

Cf. A035294, A078409, A078410.

Cf. A005408, A000009.

Sequence in context: A136275 A344002 A328170 * A007478 A014605 A365828

Adjacent sequences: A078405 A078406 A078407 * A078409 A078410 A078411

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 27 2002

Extensions

More terms from Reinhard Zumkeller, Dec 28 2002

Status

approved