A066897 -id:A066897 - cp's OEIS frontend

A066966 Total sum of even parts in all partitions of n.

0, 2, 2, 10, 12, 30, 40, 82, 110, 190, 260, 422, 570, 860, 1160, 1690, 2252, 3170, 4190, 5760, 7540, 10142, 13164, 17450, 22442, 29300, 37410, 48282, 61170, 78132, 98310, 124444, 155582, 195310, 242722, 302570, 373882, 462954, 569130, 700570, 856970

Offset: 1

Vladeta Jovovic, Jan 26 2002

nonn

Partial sums of A206436. - Omar E. Pol, Mar 17 2012
From Omar E. Pol, Apr 02 2023: (Start)
Convolution of A000041 and A146076.
Convolution of A002865 and A271342.
a(n) is also the sum of all even divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned even divisors are also all even parts of all partitions of n. (End)

			a(4) = 10 because in the partitions of 4, namely [4],[3,1],[2,2],[2,1,1],[1,1,1,1], the total sum of the even parts is 4+2+2+2 = 10.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
George E. Andrews and Mircea Merca, A further look at the sum of the parts with the same parity in the partitions of n, Journal of Combinatorial Theory, Series A, Volume 203, 105849 (2024).

Cf. A000041, A000203, A002865, A066186, A066897, A066898, A066967, A113686, A146076, A206436, A271342, A338156.

g:=sum(2*j*x^(2*j)/(1-x^(2*j)),j=1..55)/product(1-x^j,j=1..55): gser:=series(g,x=0,45): seq(coeff(gser,x^n),n=1..41);
# Emeric Deutsch, Feb 20 2006
b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+ ((i+1) mod 2)*g[1]*i]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);
# Alois P. Heinz, Mar 22 2012

max = 50; g = Sum[2*j*x^(2*j)/(1 - x^(2*j)), {j, 1, max}]/Product[1 - x^j, {j, 1, max}]; gser = Series[g, {x, 0, max}]; a[n_] := SeriesCoefficient[gser, {x, 0, n}]; Table[a[n], {n, 1, max - 1}] (* Jean-François Alcover, Jan 24 2014, after Emeric Deutsch *)
Map[Total[Select[Flatten[IntegerPartitions[#]], EvenQ]] &, Range[30]] (* Peter J. C. Moses, Mar 14 2014 *)

a(n) = 2*sum(k=1, floor(n/2), sigma(k)*numbpart(n-2*k) ); \\ Joerg Arndt, Jan 24 2014

a(n) = 2*Sum_{k=1..floor(n/2)} sigma(k)*numbpart(n-2*k).
a(n) = Sum_{k=0..n} k*A113686(n,k). - Emeric Deutsch, Feb 20 2006
G.f.: Sum_{j>=1} (2jx^(2j)/(1-x^(2j)))/Product_{j>=1}(1-x^j). - Emeric Deutsch, Feb 20 2006
a(n) = A066186(n) - A066967(n). - Omar E. Pol, Mar 10 2012
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)). - Vaclav Kotesovec, May 29 2018

More terms from Naohiro Nomoto and Sascha Kurz, Feb 07 2002

More terms from Emeric Deutsch, Feb 20 2006

A194714 Sum of all odd-indexed parts minus the sum of all even-indexed parts of all partitions of n, with the parts written in nondecreasing order.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 14, 18, 26, 32, 48, 57, 82, 102, 138, 169, 230, 278, 370, 450, 584, 709, 914, 1102, 1400, 1692, 2124, 2555, 3186, 3818, 4720, 5649, 6926, 8269, 10078, 11989, 14526, 17249, 20782, 24603, 29508, 34843, 41600, 49008, 58258, 68468, 81098

Offset: 1

Omar E. Pol, Feb 12 2012

nonn

It appears that A066897 is also another version of this sequence but with the parts written in nonincreasing order.

			a(6) = 37 - 29 = 8 because the partitions of 6 written in nondecreasing order are
.
.   6                        =  6
.   3  - 3                   =  0
.   2  - 4                   = -2
.   2  - 2  + 2              =  2
.   1  - 5                   = -4
.   1  - 2  + 3              =  2
.   1  - 1  + 4              =  4
.   1  - 1  + 2 - 2          =  0
.   1  - 1  + 1 - 3          = -2
.   1  - 1  + 1 - 1 + 2      =  2
.   1  - 1  + 1 - 1 + 1 - 1  =  0
----------------------------------
.  20 - 21 + 14 - 7 + 3 - 1  =  8

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A066897, A135010, A138121, A141285, A194853.

More terms from Alois P. Heinz, Feb 12 2012

A305123 G.f.: Sum_{k>=1} x^(2k-1)/(1+x^(2k-1)) * Product_{k>=1} 1/(1-x^k).

Original entry on oeis.org

0, 1, 0, 3, 2, 7, 6, 15, 16, 32, 36, 62, 74, 117, 142, 214, 264, 377, 468, 648, 806, 1090, 1354, 1791, 2224, 2894, 3580, 4598, 5670, 7193, 8838, 11102, 13588, 16925, 20632, 25501, 30972, 38021, 46000, 56135, 67668, 82119, 98642, 119115, 142592, 171412, 204520

Offset: 0

Vaclav Kotesovec, May 26 2018

nonn

Conjecture: a(n) is odd iff n is a term of A067567. - Peter Bala, Jan 10 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A001227, A006128, A067567, A209423, A066898, A305121, A066897.

Cf. A116676, A305124.

nmax = 60; CoefficientList[Series[Sum[x^(2*k-1)/(1+x^(2*k-1)), {k, 1, nmax}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

For n > 0, a(n) = A209423(n) - A305121(n).

a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (2^(5/2)*Pi*sqrt(n)).

A305121 G.f.: Sum_{k>=1} x^(2k)/(1+x^(2k)) * Product_{k>=1} 1/(1-x^k).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 7, 9, 14, 20, 32, 43, 63, 85, 122, 162, 221, 292, 396, 514, 680, 878, 1147, 1465, 1886, 2391, 3050, 3836, 4841, 6048, 7579, 9403, 11685, 14419, 17806, 21845, 26810, 32725, 39947, 48528, 58926, 71267, 86151, 103750, 124860, 149791, 179551

Offset: 0

Vaclav Kotesovec, May 26 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A006128, A209423, A066898, A066897, A305123.

Cf. A116680, A305122.

nmax = 60; CoefficientList[Series[Sum[x^(2*k)/(1+x^(2*k)), {k, 1, nmax}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

For n > 0, a(n) = A209423(n) - A305123(n).

a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (2^(5/2)*Pi*sqrt(n)).

A206433 Total number of odd parts in the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 3, 3, 7, 9, 15, 19, 32, 40, 60, 78, 111, 143, 200, 252, 343, 437, 576, 728, 952, 1190, 1531, 1911, 2426, 3008, 3788, 4664, 5819, 7143, 8830, 10780, 13255, 16095, 19661, 23787, 28881, 34795, 42051, 50445, 60675, 72547, 86859, 103481, 123442, 146548

Offset: 1

Omar E. Pol, Feb 12 2012

nonn

From Omar E. Pol, Apr 07 2023: (Start)
Convolution of A002865 and A001227.
a(n) is also the total number of odd divisors of the terms in the n-th row of the triangle A336811.
a(n) is also the number of odd terms in the n-th row of the triangle A207378.
a(n) is also the number of odd terms in the n-th row of the triangle A336812. (End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Partial sums give A066897.

Cf. A001227, A002865, A006128, A135010, A138121, A138137, A182703, A206434, A206435, A206436, A207378, A336811, A336812.

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+ (i mod 2)*g[1]]
      fi
    end:
a:= n-> b(n, n)[2] -b(n-1, n-1)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Mar 22 2012

b[n_, i_] := b[n, i] = Module[{f, g}, If[n==0 || i==1, {1, n}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]]+g[[1]], f[[2]]+g[[2]] + Mod[i, 2]*g[[1]]}]]; a[n_] := b[n, n][[2]]-b[n-1, n-1][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 16 2017, after Alois P. Heinz *)

More terms from Alois P. Heinz, Mar 22 2012

A207381 Total sum of the odd-indexed parts of all partitions of n.

Original entry on oeis.org

1, 3, 7, 14, 25, 45, 72, 117, 180, 275, 403, 596, 846, 1206, 1681, 2335, 3183, 4342, 5820, 7799, 10321, 13622, 17798, 23221, 30009, 38706, 49567, 63316, 80366, 101805, 128211, 161134, 201537, 251495, 312508, 387535, 478674, 590072, 724920, 888795, 1086324

Offset: 1

Omar E. Pol, Feb 17 2012

nonn

For more information see A206563.

			For n = 5, write the partitions of 5 and below write the sums of their odd-indexed parts:
.    5
.    3+2
.    4+1
.    2+2+1
.    3+1+1
.    2+1+1+1
.    1+1+1+1+1
.  ------------
.   20 + 4 + 1 = 25
The total sum of the odd-indexed parts is 25 so a(5) = 25.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A066186, A066897, A066898, A181187, A194714, A206283, A206563, A207031, A207032, A207382.

b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0$2]
    elif i<1 then [0$3]
    else g:= b(n, i-1); h:= `if`(i>n, [0$3], b(n-i, i));
         [g[1]+h[1], g[2]+h[3], g[3]+h[2]+i*h[1]]
      fi
    end:
a:= n-> b(n,n)[3]:
seq(a(n), n=1..50); # Alois P. Heinz, Mar 12 2012

b[n_, i_] := b[n, i] = Module[{g, h}, If[n == 0 , {1, 0, 0}, If[i < 1, {0, 0, 0},  g = b[n, i - 1]; h = If[i > n, {0, 0, 0}, b[n - i, i]]; {g[[1]] + h[[1]], g[[2]] + h[[3]], g[[3]] + h[[2]] + i*h[[1]]}]]]; a[n_] := b[n, n][[3]]; Table [a[n], {n, 1, 50}] (* Jean-François Alcover, Dec 09 2016 after Alois P. Heinz *)

a(n) = A066186(n) - A207382(n) = A066897(n) + A207382(n).

More terms from Alois P. Heinz, Mar 12 2012

A207382 Sum of the even-indexed parts of all partitions of n.

Original entry on oeis.org

0, 1, 2, 6, 10, 21, 33, 59, 90, 145, 213, 328, 467, 684, 959, 1361, 1866, 2588, 3490, 4741, 6311, 8422, 11067, 14579, 18941, 24630, 31703, 40788, 52019, 66315, 83891, 106034, 133182, 167045, 208397, 259637, 321895, 398498, 491295, 604725, 741579, 908008

Offset: 1

Omar E. Pol, Feb 17 2012

nonn

Also the sum of the floors of half the parts of all partitions of n, because the sum of one kind for a partition equals the sum of the other kind for the conjugate partition. Furthermore, this generalizes to taking m-th indices and dividing by m. - George Beck, Apr 15 2017

			For n = 5, write the partitions of 5 and below write the sums of their even-indexed parts:
. 5
. 3+2
. 4+1
. 2+2+1
. 3+1+1
. 2+1+1+1
. 1+1+1+1+1
------------
.   8 + 2   = 10
The sum of the even-indexed parts is 10, so a(5) = 10.
From _George Beck_, Apr 15 2017: (Start)
Alternatively, sum the floors of the parts divided by 2:
. 2
. 1+1
. 2+0
. 1+1+0
. 1+0+0
. 1+0+0+0
. 0+0+0+0+0
The sum is 10, so a(5) = 10. (End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

For more information see A206563.

Cf. A066186, A066897, A066898, A181187, A194714, A206283, A207031, A207032, A207381.

b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0$2]
    elif i<1 then [0$3]
    else g:= b(n, i-1); h:= `if`(i>n, [0$3], b(n-i, i));
         [g[1]+h[1], g[2]+h[3], g[3]+h[2]+i*h[1]]
      fi
    end:
a:= n-> b(n,n)[2]:
seq (a(n), n=1..50); # Alois P. Heinz, Mar 12 2012

b[n_, i_] := b[n, i] = Module[{g, h}, Which[n==0, {1, 0, 0}, i<1, {0, 0, 0}, True, g = b[n, i-1]; h = If[i>n, {0, 0, 0}, b[n-i, i]]; {g[[1]] + h[[1]], g[[2]] + h[[3]], g[[3]] + h[[2]] + i*h[[1]]}]]; a[n_] := b[n, n][[2]]; Table [a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *)
a[n_]:= Total@Flatten@Quotient[IntegerPartitions[n], 2];
Table [a[n], {n, 1, 50}] (* George Beck, Apr 15 2017 *)

a(n) = A066186(n) - A207381(n) = A207381(n) - A066897(n).

More terms from Alois P. Heinz, Mar 12 2012

A245840 Triangle T read by rows: T(n,k) = Total number of odd parts in all partitions of n with exactly k parts, 1 <= k <= n.

Original entry on oeis.org

1, 0, 2, 1, 1, 3, 0, 2, 2, 4, 1, 2, 4, 3, 5, 0, 4, 4, 6, 4, 6, 1, 3, 8, 7, 8, 5, 7, 0, 4, 8, 12, 10, 10, 6, 8, 1, 4, 13, 14, 17, 13, 12, 7, 9, 0, 6, 12, 22, 20, 22, 16, 14, 8, 10, 1, 5, 18, 25, 32, 27, 27, 19, 16, 9, 11

Offset: 1

L. Edson Jeffery, Aug 03 2014

			Triangle begins
1
0  2
1  1   3
0  2   2   4
1  2   4   3   5
0  4   4   6   4   6
1  3   8   7   8   5   7
0  4   8  12  10  10   6   8
1  4  13  14  17  13  12   7   9
0  6  12  22  20  22  16  14   8 10
1  5  18  25  32  27  27  19  16  9  11

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A066897 (row sums), A245841 (partial sums of row entries).

Cf. A245842, A245843.

b:= proc(n, i, k) option remember; `if`(n=0, [`if`(k=0, 1, 0), 0],
      `if`(i<1 or k=0, [0$2], ((f, g)-> f+g+[0, `if`(irem(i, 2)=1,
       g[1], 0)])(b(n, i-1, k), `if`(i>n, [0$2], b(n-i, i, k-1)))))
    end:
T:= (n, k)-> b(n$2, k)[2]:
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Aug 04 2014

Grid[Table[Sum[Count[Flatten[IntegerPartitions[n, {k}]], i], {i, 1, n, 2}], {n, 1, 11}, {k, 1, n}]]
b[n_, i_, k_] := b[n, i, k] = If[n==0, {If[k==0, 1, 0], 0}, If[i<1 || k==0, {0, 0}, Function[{f, g}, f+g+{0, If[Mod[i, 2]==1, g[[1]], 0]}][b[n, i-1, k], If[i>n, {0, 0}, b[n-i, i, k-1]]]]]; T[n_, k_] := b[n, n, k][[2]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Nov 17 2015, after Alois P. Heinz *)

T(n,k) + A245842(n,k) = A172467(n,k).

A245841 Triangle T read by rows: T(n,k) = Total number of odd parts in all partitions of n with at most k parts, 1 <= k <= n.

Original entry on oeis.org

1, 0, 2, 1, 2, 5, 0, 2, 4, 8, 1, 3, 7, 10, 15, 0, 4, 8, 14, 18, 24, 1, 4, 12, 19, 27, 32, 39, 0, 4, 12, 24, 34, 44, 50, 58, 1, 5, 18, 32, 49, 62, 74, 81, 90, 0, 6, 18, 40, 60, 82, 98, 112, 120, 130, 1, 6, 24, 49, 81, 108, 135, 154, 170, 179, 190

Offset: 1

L. Edson Jeffery, Aug 03 2014

			Triangle begins:
1
0  2
1  2   5
0  2   4   8
1  3   7  10  15
0  4   8  14  18   24
1  4  12  19  27   32   39
0  4  12  24  34   44   50   58
1  5  18  32  49   62   74   81   90
0  6  18  40  60   82   98  112  120  130
1  6  24  49  81  108  135  154  170  179  190

Alois P. Heinz, Rows n = 1..141, flattened

Partial sums of row entries of A245840.
Cf. A066897 (outer diagonal).
Cf. A245842, A245843.

b:= proc(n, i, k) option remember; `if`(n=0, [`if`(k=0, 1, 0), 0],
      `if`(i<1 or k=0, [0$2], ((f, g)-> f+g+[0, `if`(irem(i, 2)=1,
       g[1], 0)])(b(n, i-1, k), `if`(i>n, [0$2], b(n-i, i, k-1)))))
    end:
T:= proc(n, k) T(n, k):= b(n$2, k)[2]+`if`(k=1, 0, T(n, k-1)) end:
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Aug 04 2014

Grid[Table[Sum[Sum[Count[Flatten[IntegerPartitions[n, {j}]], i], {i, 1, n, 2}], {j, k}], {n, 11}, {k, n}]]
(* second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, {If[k == 0, 1, 0], 0}, If[i < 1 || k == 0, {0, 0}, Function[{f, g}, f + g + {0, If[Mod[i, 2] == 1, g[[1]], 0]}][b[n, i - 1, k], If[i > n, {0, 0}, b[n - i, i, k - 1]]]]];
T[n_, k_] := b[n, n, k][[2]];
Table[Table[T[n, k], {k, 1, n}] // Accumulate, {n, 1, 14}] // Flatten (* Jean-François Alcover, Jul 21 2016, after Alois P. Heinz *)

A245842 Triangle T read by rows: T(n,k) = Total number of even parts in all partitions of n with exactly k parts, 1 <= k <= n.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 0, 2, 2, 1, 0, 1, 2, 5, 2, 1, 0, 0, 3, 4, 5, 2, 1, 0, 1, 4, 7, 8, 5, 2, 1, 0, 0, 4, 8, 10, 8, 5, 2, 1, 0, 1, 4, 12, 14, 15, 8, 5, 2, 1, 0, 0, 5, 12, 19, 18, 15, 8, 5, 2, 1, 0, 1, 6, 18, 24, 27, 24, 15, 8, 5, 2, 1, 0

Offset: 1

L. Edson Jeffery, Aug 03 2014

Column sequences appear to converge to A066897.

			Triangle begins
0
1  0
0  1   0
1  2   1   0
0  2   2   1   0
1  2   5   2   1   0
0  3   4   5   2   1  0
1  4   7   8   5   2  1  0
0  4   8  10   8   5  2  1  0
1  4  12  14  15   8  5  2  1  0
0  5  12  19  18  15  8  5  2  1  0

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A245843 (partial sums of row entries).
Cf. A066898 (row sums), A172467.
Cf. A245840, A245841.

b:= proc(n, i, k) option remember; `if`(n=0, [`if`(k=0, 1, 0), 0],
      `if`(i<1 or k=0, [0$2], ((f, g)-> f+g+[0, `if`(irem(i, 2)=0,
       g[1], 0)])(b(n, i-1, k), `if`(i>n, [0$2], b(n-i, i, k-1)))))
    end:
T:= (n, k)-> b(n$2, k)[2]:
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Aug 04 2014

Grid[Table[Sum[Count[Flatten[IntegerPartitions[n, {k}]], i], {i, 2, n, 2}], {n, 11}, {k, n}]]
(* second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, {If[k == 0, 1, 0], 0}, If[i < 1 || k == 0, {0, 0}, Function[{f, g}, f + g + {0, If[Mod[i, 2] == 0, g[[1]], 0]}][b[n, i-1, k], If[i > n, {0, 0}, b[n-i, i, k-1]]]]];
T[n_, k_] := b[n, n, k][[2]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, May 21 2016, after Alois P. Heinz *)

T(n,k) + A245840(n,k) = A172467(n,k).

cp's OEIS Frontend

A066966 Total sum of even parts in all partitions of n.

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A194714 Sum of all odd-indexed parts minus the sum of all even-indexed parts of all partitions of n, with the parts written in nondecreasing order.

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A305123 G.f.: Sum_{k>=1} x^(2*k-1)/(1+x^(2*k-1)) * Product_{k>=1} 1/(1-x^k).

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A305121 G.f.: Sum_{k>=1} x^(2*k)/(1+x^(2*k)) * Product_{k>=1} 1/(1-x^k).

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A206433 Total number of odd parts in the last section of the set of partitions of n.

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A207381 Total sum of the odd-indexed parts of all partitions of n.

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A207382 Sum of the even-indexed parts of all partitions of n.

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Extensions

A305123 G.f.: Sum_{k>=1} x^(2k-1)/(1+x^(2k-1)) * Product_{k>=1} 1/(1-x^k).

A305121 G.f.: Sum_{k>=1} x^(2k)/(1+x^(2k)) * Product_{k>=1} 1/(1-x^k).