A104575 -id:A104575 - cp's OEIS frontend

A090794 Number of partitions of n such that the number of different parts is odd.

1, 2, 2, 3, 2, 5, 4, 9, 13, 19, 27, 43, 54, 71, 102, 124, 161, 200, 257, 319, 400, 484, 618, 761, 956, 1164, 1450, 1806, 2226, 2741, 3367, 4137, 5020, 6163, 7485, 9042, 10903, 13172, 15721, 18956, 22542, 26925, 31935, 37962, 44861, 53183, 62651

Offset: 1

Vladeta Jovovic, Feb 12 2004

			n=6 has A000041(6)=11 partitions: 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1 and 1+1+1+1+1+1 with partition sets: {6}, {1,5}, {2,4}, {1,4}, {3}, {1,2,3}, {1,3}, {2}, {1,2}, {1,2} and {1}, five of them have an odd number of elements, therefore a(6)=5.

Cf. A060177, A002134, A000005, A027193, A090794.
Cf. also A092314, A092322, A092269, A092309, A092321, A092313, A092310, A092311, A092268
Cf. A092306.

import Data.List (group)
a090794 = length . filter odd . map (length . group) . ps 1 where
   ps x 0 = [[]]
   ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Dec 19 2013

a(n) = b(n, 1, 0, 0) with b(n, i, j, f) = if iReinhard Zumkeller, Feb 19 2004
G.f.: F(x)*G(x)/2, where F(x) = 1-Product(1-2*x^i, i=1..infinity) and G(x) = 1/Product(1-x^i, i=1..infinity).
a(n) = (A000041(n)-A104575(n))/2.
G.f. A(x) equals the off-diagonal entries in the 2 X 2 matrix Product_{n >= 1} [1, x^n/(1 - x^n); x^n/(1 - x^n), 1] = [B(x), A(x); A(x), B(x)], where B(x) is the g.f. of A092306. - Peter Bala, Feb 10 2021

More terms from Reinhard Zumkeller, Feb 17 2004

Definition simplified and shortened by Jonathan Sondow, Oct 13 2013

A092306 Number of partitions of n such that the set of parts has an even number of elements.

Original entry on oeis.org

1, 0, 0, 1, 2, 5, 6, 11, 13, 17, 23, 29, 34, 47, 64, 74, 107, 136, 185, 233, 308, 392, 518, 637, 814, 1002, 1272, 1560, 1912, 2339, 2863, 3475, 4212, 5123, 6147, 7398, 8935, 10734, 12843, 15464, 18382, 22041, 26249, 31326, 37213, 44273, 52375, 62103, 73376

Offset: 0

Vladeta Jovovic, Feb 12 2004

			The partitions of five are: {{5}, {4, 1}, {3, 2}, {3, 1, 1}, {2, 2, 1}, {2, 1, 1, 1}, {1, 1, 1, 1, 1}}, The seven partitions have 1, 2, 2, 2, 2, 2 and 1 distinct parts respectively.
n=6 has A000041(6)=11 partitions: 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1 and 1+1+1+1+1+1 with partition sets: {6}, {1,5}, {2,4}, {1,4}, {3}, {1,2,3}, {1,3}, {2}, {1,2}, {1,2} and {1}, six of them have an even number of elements, therefore a(6)=6.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A060177, A002133, A027187, A090794.

import Data.List (group)
a092306 = length . filter even . map (length . group) . ps 1 where
   ps x 0 = [[]]
   ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Dec 19 2013

b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
       b(n, i-1, t) +add(b(n-i*j, i-1, 1-t), j=1..n/i)))
    end:
a:= n-> b(n, n, 1):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 29 2014

f[n_] := Count[ Mod[ Length /@ Union /@ IntegerPartitions[n], 2], 0]; Table[ f[n], {n, 0, 49}] (* Robert G. Wilson v, Feb 16 2004, updated by Jean-François Alcover, Jan 29 2014 *)

a(n) = b(n, 1, 0, 1) with b(n, i, j, f) = if iReinhard Zumkeller, Feb 19 2004
G.f.: F(x)*G(x)/2, where F(x) = 1+Product(1-2*x^i, i=1..infinity) and G(x) = 1/Product(1-x^i, i=1..infinity).
a(n) = (A000041(n)+A104575(n))/2.
G.f. A(x) equals the main diagonal entries in the 2 X 2 matrix Product_{n >= 1} [1, x^n/(1 - x^n); x^n/(1 - x^n), 1] = [A(x), B(x); B(x), A(x)], where B(x) is the g.f. of A090794. - Peter Bala, Feb 10 2021

More terms from Robert G. Wilson v, Feb 16 2004

A329157 Expansion of Product_{k>=1} (1 - Sum_{j>=1} j * x^(k*j)).

Original entry on oeis.org

1, -1, -3, -3, -4, 3, 2, 19, 21, 32, 40, 45, 16, 8, -18, -125, -164, -291, -358, -530, -588, -724, -592, -675, -358, -207, 570, 1201, 2208, 3333, 4944, 6490, 8277, 10492, 11800, 13260, 14328, 14722, 12942, 12075, 5640, 603, -10444, -21120, -39360, -55876, -83488

Offset: 0

Ilya Gutkovskiy, Nov 06 2019

sign

Convolution inverse of A329156.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A003056, A032198, A077285, A088305, A104575, A319668, A329156, A385001.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>1, b(n, i-1), 0)-
      add(b(n-i*j, min(n-i*j, i-1))*j, j=`if`(i=1, n, 1..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..46);  # Alois P. Heinz, Jul 18 2025

nmax = 46; CoefficientList[Series[Product[(1 - Sum[j x^(k j), {j, 1, nmax}]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 46; CoefficientList[Series[Product[(1 - x^k/(1 - x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^k / (1 - x^k)^2).
G.f.: exp(-Sum_{k>=1} ( Sum_{d|k} 1 / (d * (1 - x^(k/d))^(2*d)) ) * x^k).
G.f.: Product_{k>=1} (1 - x^k)^A032198(k).
G.f.: A(x) = Product_{k>=1} 1 / B(x^k), where B(x) = g.f. of A088305.
a(n) = Sum_{k=0..A003056(n)} (-1)^k * A385001(n,k). - Alois P. Heinz, Jul 18 2025

A104510 G.f.: Product_{i>=1} (1 - 2*(-x)^i)/(1 - (-x)^i)^2.

Original entry on oeis.org

0, -1, 2, -4, 4, -7, 4, -5, 0, 5, -18, 23, -46, 65, -82, 108, -132, 152, -164, 168, -144, 132, -48, -39, 212, -365, 658, -947, 1382, -1800, 2394, -2947, 3644, -4289, 5102, -5687, 6392, -6820, 7112, -7139, 6776, -5836, 4338, -2036, -1342, 5585, -11392, 18513, -27456, 37876, -51072, 65488, -82982, 101898

Offset: 1

Vladeta Jovovic, Apr 19 2005

Cf. A000712, A077285.

Cf. A104575.

gf:=product((1-2*(-x)^i)/(1-(-x)^i)^2, i=1..100): s:=series(gf, x, 100): for n from 1 to 99 do printf(`%d,`,coeff(s, x, n)) od: # James Sellers, Apr 22 2005

a(n) = Sum (k(1)-1)*(k(2)-1)*...*(k(n)-1), where the sum is taken over all (k(1), k(2), ..., k(n)) such that k(1) + 2*k(2) + ... + n*k(n) = n, k(i) >= 0, i=1..n.

G.f.: Product_{i>=1} (1 - (-x)^i)^A052823(i). - James Sellers, Apr 22 2005

More terms from James Sellers, Apr 22 2005

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A090794 Number of partitions of n such that the number of different parts is odd.

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A092306 Number of partitions of n such that the set of parts has an even number of elements.

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A329157 Expansion of Product_{k>=1} (1 - Sum_{j>=1} j * x^(k*j)).

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A104510 G.f.: Product_{i>=1} (1 - 2*(-x)^i)/(1 - (-x)^i)^2.

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