A116680 -id:A116680 - cp's OEIS frontend

A015723 Number of parts in all partitions of n into distinct parts.

1, 1, 3, 3, 5, 8, 10, 13, 18, 25, 30, 40, 49, 63, 80, 98, 119, 149, 179, 218, 266, 318, 380, 455, 541, 640, 760, 895, 1050, 1234, 1442, 1679, 1960, 2272, 2635, 3052, 3520, 4054, 4669, 5359, 6142, 7035, 8037, 9170, 10460, 11896, 13517, 15349, 17394, 19691

Offset: 1

Clark Kimberling

nonn

			The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)} with a total of 1 + 2 + 2 + 3 = 8 parts, so a(6) = 8. - _Gus Wiseman_, May 09 2019

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Arnold Knopfmacher, and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75. See s(n).
Eric Weisstein's World of Mathematics, q-Polygamma Function, q-Pochhammer Symbol.

Cf. A006128, A008289, A079499, A067619, A186545.
Column k=1 of: A210485, A213177, A327622.
Row lengths of A325537.
Cf. A022629, A066186, A066189, A325504, A325505, A325506, A325513, A325515.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1)), j=0..min(n/i, 1))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 27 2013

nn=50; Rest[CoefficientList[Series[D[Product[1+y x^i,{i,1,nn}],y]/.y->1,{x,0,nn}],x]]  (* Geoffrey Critzer, Oct 29 2012; fixed by Vaclav Kotesovec, Apr 16 2016 *)
q[n_, k_] := q[n, k] = If[nVaclav Kotesovec, Apr 16 2016 *)
Table[Length[Join@@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,1,50}] (* Gus Wiseman, May 09 2019 *)
b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0},
   Sum[{#[[1]], #[[2]] + #[[1]]*j}&@ b[n-i*j, i-1], {j, 0, Min[n/i, 1]}]]];
a[n_] := b[n, n][[2]];
Array[a, 50] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

N=66;  q='q+O('q^N); gf=sum(n=0,N, n*q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
Vec(gf) /* Joerg Arndt, Oct 20 2012 */

G.f.: sum(k>=1, x^k/(1+x^k) ) * prod(m>=1, 1+x^m ). Convolution of A048272 and A000009. - Vladeta Jovovic, Nov 26 2002
G.f.: sum(k>=1, k*x^(k*(k+1)/2)/prod(i=1..k, 1-x^i ) ). - Vladeta Jovovic, Sep 21 2005
a(n) = A238131(n)+A238132(n) = sum_{k=1..n} A048272(k)*A000009(n-k). - Mircea Merca, Feb 26 2014
a(n) = Sum_{k>=1} k*A008289(n,k). - Vaclav Kotesovec, Apr 16 2016
G.f.: -(-1; x)inf * (log(1-x) + psi_x(1 - log(-1)/log(x)))/(2*log(x)), where psi_q(z) is the q-digamma function, (a; q)_inf is the q-Pochhammer symbol, log(-1) = i*Pi. - _Vladimir Reshetnikov, Nov 21 2016
a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (2 * Pi * n^(1/4)). - Vaclav Kotesovec, May 19 2018
For n > 0, a(n) = A116676(n) + A116680(n). - Vaclav Kotesovec, May 26 2018

Extended and corrected by Naohiro Nomoto, Feb 24 2002

A116676 Number of odd parts in all partitions of n into distinct parts.

Original entry on oeis.org

0, 1, 0, 2, 2, 3, 4, 5, 8, 10, 14, 16, 22, 26, 34, 43, 54, 64, 80, 96, 116, 142, 170, 202, 242, 288, 340, 404, 474, 556, 652, 762, 886, 1034, 1198, 1389, 1606, 1852, 2132, 2454, 2814, 3224, 3690, 4214, 4804, 5478, 6228, 7072, 8028, 9094, 10290, 11635, 13134

Offset: 0

Emeric Deutsch, Feb 22 2006

nonn

a(n) = Sum(k*A116675(n,k), k>=0).

			a(9) = 10 because in the partitions of 9 into distinct parts, namely, [9], [81], [72], [6,3], [6,2,1], [5,4], [5,3,1] and [4,3,2], we have a total of 10 odd parts.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A116675, A305123, A305124.

Cf. A305082, A015723, A090867, A116680, A067588.

f:=product(1+x^j,j=1..64)*sum(x^(2*j-1)/(1+x^(2*j-1)),j=1..35): fser:=series(f,x=0,60): seq(coeff(fser,x,n),n=0..56);
# second Maple program:
b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0] elif i<1 then [0, 0]
    else f:=b(n, i-1); g:=`if`(i>n, [0, 0], b(n-i, min(n-i, i-1)));
         [f[1]+g[1], f[2]+g[2] +irem(i, 2)*g[1]]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 21 2012

b[n_, i_] := b[n, i] = Module[{f, g}, Which [n == 0, {1, 0}, i<1 , {0, 0}, True, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, Min[n-i, i-1]]]; {f[[1]] + g[[1]],       f[[2]] + g[[2]] + Mod[i, 2]*g[[1]]}]]; a[n_] := b[n, n][[2]]; Table [a[n], {n, 0, 60}] (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)

G.f.: product(1+x^j, j=1..infinity)*sum(x^(2j-1)/(1+x^(2j-1)), j=1..infinity).
For n > 0, a(n) = A015723(n) - A116680(n). - Vaclav Kotesovec, May 26 2018
a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). - Vaclav Kotesovec, May 26 2018

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)).

A305122 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, 4, 8, 16, 28, 47, 78, 126, 198, 306, 464, 694, 1024, 1490, 2146, 3061, 4322, 6052, 8408, 11592, 15872, 21592, 29192, 39242, 52468, 69788, 92376, 121716, 159664, 208569, 271372, 351732, 454228, 584546, 749720, 958472, 1221560, 1552210, 1966698

Offset: 0

Vaclav Kotesovec, May 26 2018

nonn

Convolution of A305121 and A000009.

The g.f. Sum_{k >= 1} x^(2*k)/(1 + x^(2*k)) * Product_{k >= 1} (1 + x ^k)/(1 - x^k) = Sum_{k >= 1} x^(2*k)/(1 + x^(2*k)) * Product_{k >= 1} (1 + x ^k)/(1 + x^k - 2*x^k) is congruent mod 2 to Sum_{k >= 1} x^(2*k)/(1 + x^(2*k)) = -G(-x^2), where G(x) is the g.f. of A112329. It follows that a(n) is odd iff n = 2*k^2 for some positive integer k. - Peter Bala, Jan 07 2025

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

Cf. A305102, A305101, A305104, A305105, A305124.

Cf. A116680, A112329, A305121.

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) = A305101(n) - A305124(n).

a(n) ~ exp(sqrt(n)*Pi) * log(2) / (8*Pi*sqrt(n)).

A341496 Number of partitions of n with exactly one repeated part and that part is even.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 4, 5, 6, 9, 12, 16, 20, 26, 34, 43, 53, 67, 82, 101, 124, 151, 184, 222, 267, 320, 381, 454, 539, 637, 752, 884, 1038, 1214, 1417, 1651, 1920, 2227, 2578, 2979, 3437, 3957, 4547, 5218, 5980, 6840, 7815, 8914, 10154, 11552, 13122

Offset: 0

Andrew Howroyd, Feb 13 2021

nonn

			The a(4) = 1 partition is: 2+2.
The a(5) = 1 partition is: 1+2+2.
The a(6) = 1 partition is: 2+2+2.
The a(7) = 2 partitions are: 2+2+3, 1+2+2+2.
The a(8) = 4 partitions are: 4+4, 2+2+4, 1+2+2+3, 2+2+2+2.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Cristina Ballantine and Mircea Merca, Combinatorial proofs of two theorems related to the number of even parts in all partitions of n into distinct parts, Ramanujan J., 54:1 (2021), 107-112.

Cf. A090867, A116680, A341494, A341495, A341497.

seq(n)={Vec(sum(k=1, n\4, x^(4*k)/(1 - x^(4*k)) + O(x*x^n)) * prod(k=1, n, 1 + x^k + O(x*x^n)), -(n+1))}

G.f.: (Sum_{k>=1} x^(4*k)/(1 - x^(4*k))) * Product_{k>=1} (1 + x^k).
a(n) = A090867(n) - A341497(n).
a(n) = A341497(n) - A116680(n).
a(n) = A341494(n) for even n; a(n) = A341495(n) for odd n.

A341497 Number of partitions of n with exactly one repeated part and that part is odd.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 7, 9, 13, 17, 23, 30, 39, 49, 63, 78, 98, 122, 150, 184, 225, 272, 329, 397, 475, 567, 676, 802, 948, 1121, 1317, 1545, 1810, 2112, 2460, 2863, 3319, 3842, 4442, 5123, 5897, 6782, 7780, 8913, 10200, 11648, 13285, 15136, 17214, 19555, 22191, 25143

Offset: 0

Andrew Howroyd, Feb 13 2021

			The a(2) = 1 partition is: 1+1.
The a(3) = 1 partition is: 1+1+1.
The a(4) = 2 partitions are: 1+1+2, 1+1+1+1.
The a(5) = 3 partitions are: 1+1+3, 1+1+1+2, 1+1+1+1+1.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Cristina Ballantine and Mircea Merca, Combinatorial proofs of two theorems related to the number of even parts in all partitions of n into distinct parts, Ramanujan J., 54:1 (2021), 107-112.

Cf. A067588, A090867, A116676, A116680, A341494, A341495, A341496.

seq(n)={Vec(sum(k=1, (n+2)\4, x^(4*k-2)/(1 - x^(4*k-2)) + O(x*x^n)) * prod(k=1, n, 1 + x^k + O(x*x^n)), -(n+1))}

G.f.: (Sum_{k>=1} x^(4*k-2)/(1 - x^(4*k-2))) * Product_{k>=1} (1 + x^k).
a(n) = A090867(n) - A341496(n).
a(n) = A116680(n) + A341496(n).
a(n) = A341495(n) for even n; a(n) = A341494(n) for odd n.
a(n) = (A067588(n) - A116676(n))/2. - Peter Bala, Jan 13 2025

A116679 Triangle read by rows: T(n,k) is the number of partitions of n into distinct part and having exactly k even parts (n >= 0, k >= 0).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 2, 3, 1, 2, 4, 2, 2, 5, 3, 2, 6, 4, 3, 7, 4, 1, 3, 8, 6, 1, 3, 10, 8, 1, 4, 11, 10, 2, 5, 13, 11, 3, 5, 15, 14, 4, 5, 18, 18, 5, 6, 20, 21, 7, 7, 23, 24, 9, 1, 8, 26, 29, 12, 1, 8, 30, 36, 14, 1, 9, 34, 41, 18, 2, 11, 38, 47, 23, 3, 12, 43, 55, 28, 4

Offset: 0

Emeric Deutsch, Feb 22 2006

Row n contains floor((1 + sqrt(1+4*n))/2) terms.
Row sums yield A000009.
T(n,0) = A000700(n), T(n,1) = A096911(n) for n >= 1.
Sum_{k>=0} k*T(n,k) = A116680(n).

			T(9,2)=2 because we have [6,2,1] and [4,3,2].
Triangle starts:
  1;
  1;
  0, 1;
  1, 1;
  1, 1;
  1, 2;
  1, 2, 1;
  1, 3, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000009, A000700, A096911, A116680.

g:=product((1+x^(2*j-1))*(1+t*x^(2*j)),j=1..25): gser:=simplify(series(g,x=0,38)): P[0]:=1: for n from 1 to 27 do P[n]:=sort(coeff(gser,x^n)) od: for n from 0 to 27 do seq(coeff(P[n],t,j),j=0..floor((sqrt(1+4*n)-1)/2)) od; # yields sequence in triangular form

With[{m=25}, CoefficientList[CoefficientList[Series[Product[(1+x^(2*j- 1))*(1+t*x^(2*j)), {j,1,m+2}], {x,0,m}, {t,0,m}], x], t]]//Flatten (* G. C. Greubel, Jun 07 2019 *)

G.f.: Product_{j>=1} (1+x^(2*j-1))*(1+t*x^(2*j)).

cp's OEIS Frontend

A015723 Number of parts in all partitions of n into distinct parts.

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A116676 Number of odd parts in all partitions of n into distinct parts.

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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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A305122 G.f.: Sum_{k>=1} x^(2*k)/(1+x^(2*k)) * Product_{k>=1} (1+x^k)/(1-x^k).

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A341496 Number of partitions of n with exactly one repeated part and that part is even.

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A341497 Number of partitions of n with exactly one repeated part and that part is odd.

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A116679 Triangle read by rows: T(n,k) is the number of partitions of n into distinct part and having exactly k even parts (n >= 0, k >= 0).

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

A305122 G.f.: Sum_{k>=1} x^(2k)/(1+x^(2k)) * Product_{k>=1} (1+x^k)/(1-x^k).