A035294 Number of ways to partition 2n into distinct positive integers.
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
Offset: 0
Examples
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 + ...
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
- N. J. A. Sloane, Transforms
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Crossrefs
Programs
-
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
-
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-François Alcover, Aug 30 2016, after Alois P. Heinz *) -
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 */
Formula
a(n) = A000009(2*n). - Michael Somos, Mar 03 2003
Expansion of Sum_{n >= 0} q^n / Product_{k = 1..2*n} (1 - q^k).
a(n) = T(2*n, 0), T as defined in A026835.
G.f.: Product_{i >= 0} ((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). - 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
G.f.: 1/(1 - x)*Sum_{n>=0} x^floor((3*n+1)/2)/Product_{k = 1..n} (1 - x^k). - Peter Bala, Feb 04 2021
G.f.: Product_{n >= 1} (1 - q^(8*n))*(1 + q^(8*n-3))*(1 + q^(8*n-5))/(1 - q^n). - Peter Bala, Dec 30 2024
Comments