A116676 -id:A116676 - 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

A116680 Number of even parts in all partitions of n into distinct parts.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 4, 5, 5, 8, 11, 14, 18, 23, 29, 37, 44, 55, 69, 83, 102, 124, 148, 178, 213, 253, 300, 356, 421, 494, 582, 680, 793, 926, 1074, 1246, 1446, 1668, 1922, 2215, 2545, 2918, 3345, 3823, 4366, 4982, 5668, 6445, 7321, 8300, 9401, 10639, 12021, 13566

Offset: 0

Emeric Deutsch, Feb 22 2006

nonn

			a(9)=8 because in the partitions of 9 into distinct parts, namely, [9], [8,1], [7,2], [6,3], [6,2,1], [5,4], [5,3,1], and [4,3,2], we have a total of 8 even parts. [edited by _Rishi Advani_, Jun 07 2019]

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
D. Herden, M. R. Sepanski, J. Stanfill, C. C. Hammon, J. Henningsen, H. Ickes, J. M. Menendez, T. Poe, I. Ruiz, and E. L. Smith, Counting the parts divisible by k in all the partitions of n whose parts have multiplicity less than k, arXiv:2010.02788 [math.CO], 2020. See also Integers (2022) Vol. 22, #A49.
Runqiao Li and Andrew Y. Z. Wang, On the combinatorics of the number of even parts in all partitions with distinct parts, The Raman. J. 56 (2021) 712-727

Cf. A116679, A305121, A305122.

Cf. A305082, A015723, A090867, A067588, A116676.

m:=25; R:=PowerSeriesRing(Integers(), 3*m); [0,0] cat Coefficients(R!( (&*[1+x^j: j in [1..4*m]])*(&+[x^(2*k)/(1+x^(2*k)): k in [1..2*m]]) )); // G. C. Greubel, Jun 07 2019

f:=product(1+x^j,j=1..70)*sum(x^(2*j)/(1+x^(2*j)),j=1..40): fser:=series(f,x=0,65): seq(coeff(fser,x,n),n=0..60);
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 p+`if`(i::odd, 0, [0, p[1]]))(b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..60);  # Alois P. Heinz, May 24 2022

With[{m = 25}, CoefficientList[Series[Product[1+x^j, {j,1,4*m}]* Sum[x^(2*k)/(1+x^(2*k)), {k,1,2*m}], {x,0,3*m}], x]] (* G. C. Greubel, Jun 07 2019 *)

my(m=25); my(x='x+O('x^(3*m))); concat([0, 0], Vec( prod(j=1, 4*m, 1+x^j)*sum(k=1, 2*m, x^(2*k)/(1+x^(2*k))) )) \\ G. C. Greubel, Jun 07 2019

m = 25
R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(3*m)
s = product(1+x^j for j in (1..4*m))*sum(x^(2*k)/(1+x^(2*k)) for k in (1..2*m))
[0, 0] + s.coefficients() # G. C. Greubel, Jun 07 2019

a(n) = Sum_{k >= 0} k*A116679(n,k).
G.f.: (Product_{j >= 1} (1+x^j)) * (Sum_{k >= 1} x^(2*k)/(1+x^(2*k))).
For n > 0, a(n) = A015723(n) - A116676(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

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)).

A305124 G.f.: Sum_{k>=1} x^(2k-1)/(1+x^(2k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

Original entry on oeis.org

0, 1, 1, 4, 7, 14, 24, 42, 69, 113, 178, 276, 420, 630, 930, 1360, 1963, 2804, 3969, 5568, 7746, 10700, 14672, 19986, 27060, 36423, 48754, 64928, 86038, 113478, 149012, 194842, 253737, 329172, 425452, 547952, 703343, 899858, 1147680, 1459364, 1850310, 2339432

Offset: 0

Vaclav Kotesovec, May 26 2018

nonn

Convolution of A305123 and A000009.

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

Cf. A305102, A305101, A305104, A305122, A305105.

Cf. A116676, A305123.

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

a(n) = A305101(n) - A305122(n).

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

A116675 Triangle read by rows: T(n,k) is the number of partitions of n into distinct part and having exactly k odd parts (n>=0, k>=0).

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 3, 0, 2, 0, 2, 0, 5, 0, 2, 0, 4, 0, 7, 0, 1, 3, 0, 7, 0, 0, 10, 0, 2, 4, 0, 11, 0, 0, 14, 0, 4, 5, 0, 17, 0, 0, 19, 0, 8, 6, 0, 25, 0, 1, 0, 25, 0, 13, 0, 8, 0, 36, 0, 2, 0, 33, 0, 21, 0, 10, 0, 50, 0, 4, 0, 43, 0, 33, 0, 12, 0, 69, 0, 8, 0, 55, 0, 49, 0, 15, 0, 93, 0, 14

Offset: 0

Emeric Deutsch, Feb 22 2006

Row n contains 1+floor(sqrt(n)) terms (at the end of certain rows there is an extra 0). Row sums yield A000009. T(n,0) = A035457(n) (n>=1); T(2n,0) = A000009(n), T(2n-1,0)=0. T(2n,1)=0, T(2n-1,1) = A036469(n). Sum(k*T(n,k), k>=0) = A116676(n).

			T(8,2) = 4 because we have [7,1], [5,3], [5,2,1] and [4,3,1] ([8] and [6,2] do not qualify).
Triangle starts:
1;
0, 1;
1, 0;
0, 2;
1, 0, 1;
0, 3, 0;

Alois P. Heinz, Rows n = 0..750, flattened

Cf. A000009, A035457, A036469, A116676.

g:=product((1+t*x^(2*j-1))*(1+x^(2*j)),j=1..25): gser:=simplify(series(g,x=0,38)): P[0]:=1: for n from 1 to 26 do P[n]:=sort(coeff(gser,x^n)) od: for n from 0 to 26 do seq(coeff(P[n],t,j),j=0..floor(sqrt(n))) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)->
      x+y, b(n, i-1), `if`(i>n, [], [`if`(irem(i,2)=0, [][], 0),
      b(n-i, i-1)[]]), 0)))
    end:
T:= proc(n) local l; l:= b(n, n); l[], 0$(1+floor(sqrt(n))-nops(l)) end:
seq (T(n), n=0..30);  # Alois P. Heinz, Nov 21 2012

rows = 25; coes = CoefficientList[Product[(1+t*x^(2j-1))(1+x^(2j)), {j, 1, rows}], {x, t}][[1 ;; rows]]; MapIndexed[Take[#1, Floor[Sqrt[#2[[1]]-1]]+1]&, coes] // Flatten (* Jean-François Alcover, May 13 2015 *)

G.f.: product((1+tx^(2j-1))(1+x^(2j)), j=1..infinity).

A116681 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, in which the sum of the odd parts is k (n>=0, 0<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 3, 0, 0, 0, 2, 0, 1, 0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 1, 0, 2, 0, 2, 4, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 4, 0, 3, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3

Offset: 0

Emeric Deutsch, Feb 22 2006

Row sums yield A000009. T(2n,0)=A000009(n), T(2n-1,0)=0. T(2n,1)=0, T(2n+1,1)=A000009(n), T(n,2)=0. T(n,n)=A000700(n). Sum(k*T(n,k), k=0..n)=A116682(n).

			T(10,4)=2 because we have [6,3,1] and [4,3,2,1].
Triangle starts:
1;
0,1;
1,0,0;
0,1,0,1;
1,0,0,0,1;
0,1,0,1,0,1;

Cf. A000009, A035457, A036469, A116676.

g:=product((1+(t*x)^(2*j-1))*(1+x^(2*j)),j=1..30): gser:=simplify(series(g,x=0,20)): P[0]:=1: for n from 1 to 15 do P[n]:=sort(coeff(gser,x^n)) od: for n from 0 to 15 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form

G.f.=product((1+tx^(2j-1))(1+x^(2j)), j=1..infinity).

cp's OEIS Frontend

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

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A116680 Number of even parts in all partitions of n into distinct parts.

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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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A305124 G.f.: Sum_{k>=1} x^(2*k-1)/(1+x^(2*k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

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

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A116675 Triangle read by rows: T(n,k) is the number of partitions of n into distinct part and having exactly k odd parts (n>=0, k>=0).

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A116681 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, in which the sum of the odd parts is k (n>=0, 0<=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).

A305124 G.f.: Sum_{k>=1} x^(2k-1)/(1+x^(2k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).