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A035294 Number of ways to partition 2n into distinct positive integers. 17
1, 1, 2, 4, 6, 10, 15, 22, 32, 46, 64, 89, 122, 165, 222, 296, 390, 512, 668, 864, 1113, 1426, 1816, 2304, 2910, 3658, 4582, 5718, 7108, 8808, 10880, 13394, 16444, 20132, 24576, 29927, 36352, 44046, 53250, 64234, 77312, 92864, 111322, 133184, 159046 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also, number of partitions of 2n into odd numbers. - Vladeta Jovovic, Aug 17 2004

This sequence was originally defined as the expansion of sum ( q^n / product( 1-q^k, k=1..2*n), n=0..inf ). The present definition is due to Reinhard Zumkeller. Michael Somos points out that the equivalence of the two definitions follows from Andrews, page 19.

Also, number of partitions of 2n with max descent 1 and last part 1. - Wouter Meeussen, Mar 31 2013

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..1000

N. J. A. Sloane, Transforms

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

a(n) = A000009(2*n). - Michael Somos, Mar 03 2003

Expansion of sum ( q^n / product( 1-q^k, k=1..2*n), n=0..inf ).

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

G.f.: Product((1 + x^(8 * i + 1)) * (1 + x^(8 * i + 2))^2 * (1 + x^(8 * i + 3))^2 * (1 + x^(8 * i + 4))^3 * (1 + x^(8 * i + 5))^2 * (1 + x^(8 * i + 6))^2 * (1 + x^(8 * i + 7)) * (1 + x^(8 * i + 8))^3, i=0..infinity). - Vladeta Jovovic, Oct 10 2004

G.f.: (Sum_{k>=0} x^A074378(k)) / (Product_{k>0} (1 - x^k)) = f( x^3, x^5) / f(-x, -x^2) where f(, ) is Ramanujan's general theta function. - Michael Somos, Nov 01 2005

Euler transform of period 16 sequence [1, 1, 2, 1, 2, 0, 1, 0, 1, 0, 2, 1, 2, 1, 1, 0, ...]. - Michael Somos, Dec 17 2002

a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(11/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2015

a(n) = A000041(n) + A282893(n). - Michael Somos, Feb 24 2017

Convolution with A000041 is A058696. - Michael Somos, Feb 24 2017

Convolution with A097451 is A262987. - Michael Somos, Feb 24 2017

EXAMPLE

a(4)=6 [8=7+1=6+2=5+3=5+2+1=4+3+1=2*4].

G.f. = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 + 10*x^5 + 15*x^6 + 22*x^7 + 46*x^9 + ...

G.f. = q + q^49 + 2*q^97 + 4*q^145 + 6*q^193 + 10*q^241 + 15*q^289 + ...

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1,

     `if`(i<1, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i))))

    end:

a:= n-> b(2*n, 2*n-1):

seq(a(n), n=0..50);  # Alois P. Heinz, Feb 11 2015

MATHEMATICA

Table[Count[IntegerPartitions[2 n], q_ /; Union[q] == Sort[q]], {n, 16}];

Table[Count[IntegerPartitions[2 n], q_ /; Count[q, _?EvenQ] == 0], {n, 16}];

Table[Count[IntegerPartitions[2 n], q_ /; Last[q] == 1 && Max[q - PadRight[Rest[q], Length[q]]] <= 1 ], {n, 16}];

(* Wouter Meeussen, Mar 31 2013 *)

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

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3, x^8] QPochhammer[ -x^5, x^8] QPochhammer[ x^8] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, May 06 2015 *)

nmax=60; CoefficientList[Series[Product[(1+x^(8*k+1)) * (1+x^(8*k+2))^2 * (1+x^(8*k+3))^2 * (1+x^(8*k+4))^3 * (1+x^(8*k+5))^2 * (1+x^(8*k+6))^2 * (1+x^(8*k+7)) * (1+x^(8*k+8))^3, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2015 *)

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-2] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[2n, 2n-1]; Table[a[n], {n, 0, 50}] (* Jean-Fran├žois Alcover, Aug 30 2016, after Alois P. Heinz *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, n*=2; A = x * O(x^n); polcoeff( eta(x^2 + A) / eta(x + A), n))}; /* Michael Somos, Nov 01 2005 */

(Haskell)

import Data.MemoCombinators (memo2, integral)

a035294 n = a035294_list !! n

a035294_list = f 1 where

   f x = (p' 1 (x - 1)) : 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. A000009, A000041, A058686, A262987, A282893.

Cf. A078408, A078406, A078407.

Cf. A079122, A079126, A079124, A079125, A067953.

Cf. A005408.

Sequence in context: A175826 A073470 A086182 * A073818 A239288 A143184

Adjacent sequences:  A035291 A035292 A035293 * A035295 A035296 A035297

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Bill Gosper

STATUS

approved

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Last modified May 25 01:01 EDT 2017. Contains 287008 sequences.