A078406 -id:A078406 - cp's OEIS frontend

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

N. J. A. Sloane, Bill Gosper

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

			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 + ...

G. E. Andrews, The Theory of Partitions, Cambridge University Press, 1998, p. 19.

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

Cf. A000009, A000041, A058686, A262987, A282893.
Cf. A078408, A078406, A078407.
Cf. A079122, A079126, A079124, A079125, A067953.
Cf. A005408.

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

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

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 *)

{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 */

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
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
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

A323891 a(n) is the number of partitions of 72*n + 42 into 10 odd squares.

Original entry on oeis.org

2, 9, 22, 41, 68, 106, 154, 212, 285, 368, 477, 598, 741, 898, 1076, 1286, 1524, 1785, 2068, 2379, 2741, 3131, 3554, 4002, 4497, 5044, 5644, 6274, 6939, 7653, 8445, 9295, 10186, 11117, 12113, 13192, 14355, 15556, 16807, 18147, 19570, 21089, 22673, 24300, 26029, 27865, 29821, 31822, 33894, 36088

Offset: 0

Marius A. Burtea, Feb 12 2019

nonn

			For n=0, 72*0+42 = 42 = 25+9+1+1+1+1+1+1+1+1 = 9+9+9+9+1+1+1+1+1+1, so a(0)=2.
For n=1, 72*1+42 = 114 = 81+25+1+1+1+1+1+1+1+1 = 81+9+9+9+1+1+1+1+1+1 = 49+49+9+1+1+1+1+1+1+1 = 49+25+25+9+1+1+1+1+1+1 = 49+25+9+9+9+9+1+1+1+1 = 49+9+9+9+9+9+9+9+1+1 = 25+25+25+25+9+1+1+1+1+1 = 25+25+25+9+9+9+9+1+1+1 = 25+25+9+9+9+9+9+9+9+1, so a(1)=9.

Laurențiu Panaitopol, Alexandru Gica, Arithmetic problems and number theory, Ed. Gil, Zalău, (2006), ch. 14, p. 85, pr. 32. (in Romanian).

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A000041, A001156, A016754, A025425, A025434, A033461, A035294, A078406, A090677, A167661, A167700.

[#RestrictedPartitions(72*n+42, 10, {(2*d+1)^2:d in [0..100]}): n in [0..100]];

S:= proc(n, k, m)
   option remember;
   local p,j;
   if k = 0 then if n = 0 then return 1 else return 0 fi
   elif m < 1 then return 0
   elif n < k then return 0
   elif n > k*m^2 then return 0
   fi;
   if m^2 > n then
     p:= floor(sqrt(n));
     if p::even then p:= p-1 fi;
     return procname(n, k, p)
   fi;
   add(procname(n-j*m^2,k-j,m-2), j=0..n/m^2)
end proc:
seq(S(72*n+42, 10, 72*n+42), n=0..100); # Robert Israel, Feb 24 2019

a[n_] := IntegerPartitions[72n+42, {10}, Select[ Range[1, 72n+42, 2], IntegerQ@Sqrt@#&]] // Length;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 19 2022 *)

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A035294 Number of ways to partition 2n into distinct positive integers.

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A078407 Number of ways to partition 4*n+2 into distinct positive integers.

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A323891 a(n) is the number of partitions of 72*n + 42 into 10 odd squares.

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