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A078408 Number of ways to partition 2n+1 into distinct positive integers. 14
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 (list; graph; refs; listen; history; text; internal format)
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

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

M. 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

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 + ...

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: A001524 A280278 A136275 * A007478 A014605 A232477

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

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Last modified August 18 14:09 EDT 2017. Contains 290720 sequences.