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A035294
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Number of ways to partition 2n into distinct positive integers.
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24
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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G. E. Andrews, The Theory of Partitions, Cambridge University Press, 1998, p. 19.
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LINKS
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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
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FORMULA
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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 A026835.
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
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
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EXAMPLE
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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 + ...
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000009, A000041, A058686, A262987, A282893.
Cf. A078408, A078406, A078407.
Cf. A079122, A079126, A079124, A079125, A067953.
Cf. A005408.
Sequence in context: A304630 A073470 A086182 * A073818 A342091 A239288
Adjacent sequences: A035291 A035292 A035293 * A035295 A035296 A035297
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Bill Gosper
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STATUS
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approved
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