A066898 -id:A066898 - cp's OEIS frontend

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

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)).

A206434 Total number of even parts in the last section of the set of partitions of n.

Original entry on oeis.org

0, 1, 0, 3, 1, 6, 4, 13, 10, 24, 23, 46, 46, 81, 88, 143, 159, 242, 278, 404, 470, 657, 776, 1057, 1251, 1663, 1984, 2587, 3089, 3967, 4742, 6012, 7184, 9001, 10753, 13351, 15917, 19594, 23335, 28514, 33883, 41140, 48787, 58894, 69691, 83680, 98809, 118101

Offset: 1

Omar E. Pol, Feb 12 2012

nonn

From Omar E. Pol, Apr 07 2023: (Start)
Convolution of A002865 and A183063.
a(n) is also the total number of even divisors of the terms in the n-th row of the triangle A336811.
a(n) is also the number of even terms in the n-th row of the triangle A207378.
a(n) is also the number of even 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 A066898.

Cf. A002865, A006128, A135010, A138121, A138137, A182703, A183063, A206433, A206435, A206436, A207378, A336811, A336812.

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]]
      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, 0}, 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+1, 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 *)

G.f.: (Sum_{i>0} (x^(2*i)-x^(2*i+1))/(1-x^(2*i)))/Product_{i>0} (1-x^i). - Alois P. Heinz, Mar 23 2012

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

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).

A245843 Triangle T read by rows: T(n,k) = Total number of even parts in all partitions of n with at most k parts, 1 <= k <= n.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 3, 4, 4, 0, 2, 4, 5, 5, 1, 3, 8, 10, 11, 11, 0, 3, 7, 12, 14, 15, 15, 1, 5, 12, 20, 25, 27, 28, 28, 0, 4, 12, 22, 30, 35, 37, 38, 38, 1, 5, 17, 31, 46, 54, 59, 61, 62, 62, 0, 5, 17, 36, 54, 69, 77, 82, 84, 85, 85

Offset: 1

L. Edson Jeffery, Aug 03 2014

			Triangle begins:
0
1  1
0  1   1
1  3   4   4
0  2   4   5   5
1  3   8  10  11  11
0  3   7  12  14  15  15
1  5  12  20  25  27  28  28
0  4  12  22  30  35  37  38  38
1  5  17  31  46  54  59  61  62  62
0  5  17  36  54  69  77  82  84  85  85

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

Partial sums of row entries of A245842.
Cf. A066898 (outer diagonal).
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:= 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, 2, 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] == 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]] + If[k == 1, 0, T[n, k-1]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, May 21 2016, after Alois P. Heinz *)

A305104 G.f.: Sum_{k>=1} x^(2k)/(1-x^(2k)) * Product_{k>=1} (1+x^k)/(1-x^k).

Original entry on oeis.org

0, 0, 1, 2, 6, 12, 24, 44, 79, 134, 222, 358, 566, 876, 1334, 2000, 2960, 4326, 6253, 8946, 12680, 17816, 24832, 34352, 47192, 64404, 87354, 117796, 157976, 210764, 279812, 369744, 486413, 637188, 831324, 1080420, 1398968, 1805012, 2320992, 2974728, 3800618

Offset: 0

Vaclav Kotesovec, May 25 2018

nonn

Convolution A066898 of and A000009.

Convolution A090867 of and A000041.

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

Cf. A015128, A066898, A090867, A305102, A305105.

nmax = 50; CoefficientList[Series[Sum[x^(2*k)/(1-x^(2*k)), {k, 1, nmax}] * Product[(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ (2*gamma + log(n/Pi^2)) * exp(Pi*sqrt(n)) / (16*Pi*sqrt(n)), where gamma is the Euler-Mascheroni constant A001620.

A325771 Rectangular array: row n shows the number of parts in all partitions of n that are == k (mod 2), for k = 0, 1.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 4, 8, 5, 15, 11, 24, 15, 39, 28, 58, 38, 90, 62, 130, 85, 190, 131, 268, 177, 379, 258, 522, 346, 722, 489, 974, 648, 1317, 890, 1754, 1168, 2330, 1572, 3058, 2042, 4010, 2699, 5200, 3475, 6731, 4532, 8642, 5783, 11068, 7446, 14076, 9430

Offset: 1

Clark Kimberling, Jun 05 2019

Row n partitions A006128 into 2 parts, r(n,0) + r(n,1) = p(n) = A006128(n). What is the limiting behavior of r(n,0)/p(n)?

			First 15 rows:
    0    1
    1    2
    1    5
    4    8
    5   15
   11   24
   15   39
   28   58
   38   90
   62  130
   85  190
  131  268
  177  379
  258  522
  346  722

Clark Kimberling, Table of n, a(n) for n = 1..100

Cf. A006128, A066898, A066897, A325772, A325773, A325774.

f[n_] := Mod[Flatten[IntegerPartitions[n]], 2];
Table[Count[f[n], k], {n, 1, 40}, {k, 0, 1}]  (* A325771 array *)
Flatten[%] (* A325771 sequence *)

(row n) = (A066898(n), A066897(n)).

A362059 Total number of even divisors of all positive integers <= n.

Original entry on oeis.org

0, 1, 1, 3, 3, 5, 5, 8, 8, 10, 10, 14, 14, 16, 16, 20, 20, 23, 23, 27, 27, 29, 29, 35, 35, 37, 37, 41, 41, 45, 45, 50, 50, 52, 52, 58, 58, 60, 60, 66, 66, 70, 70, 74, 74, 76, 76, 84, 84, 87, 87, 91, 91, 95, 95, 101, 101, 103, 103, 111, 111, 113, 113, 119, 119, 123, 123, 127, 127, 131, 131, 140, 140, 142, 142, 146

Offset: 1

Omar E. Pol, Apr 06 2023

Convolved with A002865 gives A066898.

Partial sums of A183063.

Cf. A000005, A002865, A006218, A066898, A060831, A271342.

d[n_] := (e = IntegerExponent[n, 2]) * DivisorSigma[0, n/2^e]; Accumulate@ Array[d, 100] (* Amiram Eldar, Apr 07 2023 *)

from math import isqrt
def A362059(n): return -(q:=isqrt(m:=n>>1))**2+(sum(m//k for k in range(1, q+1))<<1) # Chai Wah Wu, Apr 26 2023

a(2n-1) = A006218(n-1), n >= 1.

a(2n) = A006218(n), n >= 1.

cp's OEIS Frontend

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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A206434 Total number of even 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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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.

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A245843 Triangle T read by rows: T(n,k) = Total number of even parts in all partitions of n with at most k parts, 1 <= k <= n.

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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).

A305104 G.f.: Sum_{k>=1} x^(2k)/(1-x^(2k)) * Product_{k>=1} (1+x^k)/(1-x^k).