A079499 -id:A079499 - 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

A067619 Total number of parts in all self-conjugate partitions of n. Also, sum of largest parts of all self-conjugate partitions of n.

Original entry on oeis.org

0, 1, 0, 2, 2, 3, 3, 4, 7, 8, 9, 10, 15, 16, 18, 23, 30, 32, 35, 42, 51, 59, 63, 73, 89, 100, 106, 125, 145, 160, 174, 198, 229, 255, 274, 310, 355, 388, 420, 472, 534, 582, 631, 701, 784, 859, 928, 1021, 1144, 1243, 1338, 1475, 1630, 1767, 1909, 2089, 2299

Offset: 0

Naohiro Nomoto, Feb 01 2002

T. D. Noe, Table of n, a(n) for n = 0..1000
Arnold Knopfmacher and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.

Cf. A000700, A000701, A006128, A015723, A046682, A079499, A158691.

CoefficientList[Series[Sum[n*q^(2n-1)*Product[1+q^k, {k, 1, 2n-3, 2}], {n, 1, 30}], {q, 0, 60}], q]

G.f.: A(q) = Sum_{n >= 1} n*q^(2*n-1)*(1+q)*(1+q^3)*...*(1+q^(2*n-3)).
From Peter Bala, Aug 20 2017: (Start)
Let F(q) = Product_{i >= 1} (1 + q^(2*i-1)). Then A(q) = Sum_{n >= 0} ( F(q) - Product_{i = 1..n} (1 + q^(2*i-1)) ).
It follows that the above sum A(q) satisfies -A(q-1) = 1 + q + 3*q^2 + 12*q^3 + 61*q^4 + ..., the g.f. for A158691, row-Fishburn matrices of size n. (End)

Edited by Dean Hickerson, Feb 11 2002

A238131 Number of parts in all partitions of n into odd number of distinct parts.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 4, 4, 7, 10, 13, 16, 22, 25, 31, 42, 48, 59, 73, 89, 108, 132, 156, 190, 227, 271, 318, 380, 449, 526, 618, 722, 841, 980, 1138, 1321, 1526, 1760, 2028, 2333, 2683, 3070, 3517, 4017, 4584, 5228, 5948, 6757, 7673, 8696, 9845, 11132, 12577

Offset: 0

Mircea Merca, Feb 18 2014

nonn

			a(8)=7 because the partitions of 8 into odd number of distinct parts are: 8, 5+2+1 and 4+3+1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Alois P. Heinz)
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, function s_o(n).
Eric Weisstein's World of Mathematics, q-Polygamma Function, q-Pochhammer Symbol.

Cf. A000005, A001318, A015723, A110654, A235963, A079499.

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, (p->
      [p[2], p[1], p[4]+p[2], p[3]+p[1]])(b(n-i, i-1)))))
    end:
a:= n-> b(n$2)[4]:
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 27 2015

max = 50; s = (1/2)*Product[1+x^k, {k, 1, max}]*Sum[x^k/(1+x^k), {k, 1, max}] + (1/2)*Product[1-x^k, {k, 1, max}]*Sum[x^k/(1-x^k), {k, 1, max}] + O[x]^(max+1); CoefficientList[s, x] (* Jean-François Alcover, Dec 27 2015 *)

a(n) = (1/2)*A015723(n)+(1/2)*sum{k=0..A235963(n)-1, (-1)^A110654(k)*A000005(n-A001318(k))}.
G.f.: (1/2)*prod(k>=1, 1+x^k ) * sum(k>=1, x^k/(1+x^k) ) + (1/2)*prod(k>=1, 1-x^k) * sum(k>=1, x^k/(1-x^k) ).
G.f.: (2 * (x; x)inf * (log(1-x) + psi_x(1)) - (-1; x)_inf * (log(1-x) + psi_x(1-log(-1)/log(x))))/(4*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)) / (4*Pi*n^(1/4)). - Vaclav Kotesovec, May 27 2018

A116422 Triangle read by rows: T(n,k) is the number of self-conjugate partitions of n having Durfee square of size k (n>=1; 1<=k<=floor(sqrt(n))).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 3, 0, 1, 0, 3, 0, 4, 0, 1, 1, 0, 4, 0, 0, 4, 0, 1, 1, 0, 5, 0, 0, 5, 0, 2, 1, 0, 7, 0, 0, 5, 0, 3, 1, 0, 8, 0, 0, 6, 0, 5, 1, 0, 10, 0, 1, 0, 6, 0, 6, 0, 1, 0, 12, 0, 1, 0, 7, 0, 9, 0, 1, 0, 14, 0, 2, 0, 7, 0, 11, 0, 1, 0, 16

Offset: 1

Emeric Deutsch, Feb 14 2006

Row n contains floor(sqrt(n)) terms (0's are possible even at the end of the rows). Row sums yield A000700. Sum_{k=1..floor(sqrt(n))} k*T(n,k) = A079499(n).

Also, number of partitions of n into k distinct odd parts. Example: T(13,3)=2 because we have [9,3,1] and [7,5,1]. - Emeric Deutsch, Feb 24 2006

			T(13,3)=2 because we have [5,3,3,1,1] and [4,4,3,2] (there is one more self-conjugate partition of 13, namely [7,1,1,1,1,1,1], having Durfee square of size 1).
Triangle starts:
1;
0;
1;
0,1;
1,0;
0,1;
1,0;
0,2;
1,0,1;
0,2,0;

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).

Álvar Ibeas, First 400 rows, flattened

Cf. A000700, A079499.

g:=sum(t^k*q^(k^2)/product(1-q^(2*i),i=1..k),k=1..15): gser:=simplify(series(g,q=0,40)): for n from 1 to 33 do P[n]:=coeff(gser,q^n) od: for n from 1 to 33 do row[n]:=seq(coeff(P[n],t^j),j=1..floor(sqrt(n))) od; # yields sequence in triangular form

rows = 31; jmax = Floor[Sqrt[rows]]; T[n_, k_] := SeriesCoefficient[Sum[ t^j*x^(j^2)/Product[1-x^(2i), {i, 1, j}], {j, 1, jmax}], {x, 0, n}, {t, 0, k}]; Table[T[n, k], {n, 1, rows}, {k, 1, Floor[Sqrt[n]]}] // Flatten (* Jean-François Alcover, Jul 16 2017 *)

G.f.: Sum_{k=1..infinity} (t^k*x^(k^2))/Product_{i=1..k} 1-x^(2*i).
G.f.: -1 + Product_{j=1..infinity} 1+t*x^(2*j-1). - Emeric Deutsch, Feb 24 2006
T(n, k) = T(n-2*k, k) + T(n-2*k+1, k-1). If n+k is even, T(n, k) = A008284((n-k^2)/2 + k, k) = A072233((n-k^2)/2, k); 0 otherwise. - Álvar Ibeas, Jul 27 2020

A097936 Total number of parts in all compositions of n into distinct odd parts.

Original entry on oeis.org

1, 0, 1, 4, 1, 4, 1, 8, 19, 8, 19, 12, 37, 12, 55, 112, 73, 112, 91, 212, 127, 308, 145, 504, 781, 600, 817, 892, 1453, 1084, 2089, 1472, 3343, 1760, 4579, 6564, 6433, 6948, 8287, 11944, 11341, 16744, 14395, 26156, 18667, 35468, 22921, 53712, 64273, 67440

Offset: 1

Vladeta Jovovic, Sep 05 2004

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

Cf. A097910, A079499, A032021.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(n>(i+1)^2/4, [][], zip((x, y)->x+y, [b(n, i-2)],
      `if`(i>n, [], [0, b(n-i, i-2)]), 0)[]))
    end:
a:= proc(n) option remember; local l; l:=[b(n, n-1+irem(n,2))];
      add(i*l[i+1]*i!, i=1..nops(l)-1)
    end:
seq (a(n), n=1..60);  # Alois P. Heinz, Nov 20 2012

Drop[ CoefficientList[ Series[Sum[k*k!*x^k^2/Product[1 - x^(2j), {j, 1, k}], {k, 1, 55}], {x, 0, 50}], x], 1] (* Robert G. Wilson v, Sep 08 2004 *)

Sum_{k>0} (k*k!*x^(k^2)/Product_{j=1..k} (1-x^(2*j))).

More terms from Robert G. Wilson v, Sep 08 2004

cp's OEIS Frontend

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

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A067619 Total number of parts in all self-conjugate partitions of n. Also, sum of largest parts of all self-conjugate partitions of n.

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A238131 Number of parts in all partitions of n into odd number of distinct parts.

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A116422 Triangle read by rows: T(n,k) is the number of self-conjugate partitions of n having Durfee square of size k (n>=1; 1<=k<=floor(sqrt(n))).

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A097936 Total number of parts in all compositions of n into distinct odd parts.

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