A238131 -id:A238131 - cp's OEIS frontend

A238133 Difference between A238131(n) and A238132(n).

0, 1, 1, -1, -1, -3, 0, -2, 1, 2, 1, 2, 4, 1, -1, 4, -2, -1, -3, -1, -2, -2, -6, 0, -1, 1, -4, 0, 3, 2, 2, 2, 3, 0, 4, 7, 0, 0, 2, -3, 7, -2, -1, -3, -2, -4, 0, -3, -3, -2, -1, -10, -1, 0, 1, -1, 0, -6, 2, 2, 0, 4, 3, 4, 0, 2, 4, 3, 0, 5, 8, 2, 0, 1, -1, 1, -3

Offset: 0

Mircea Merca, Feb 18 2014

Difference between the number of parts in all partitions of n into odd number of distinct parts and the number of parts in all partitions of n into even number of distinct parts.

The convolution of A000005 and A010815.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Mircea Merca, A new look on the generating function for the number of divisors, Journal of Number Theory, Volume 149, April 2015, Pages 57-69.
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, difference s_o(n)-s_e(n).
Eric Weisstein's World of Mathematics, q-Polygamma Function, q-Pochhammer Symbol.

Cf. A000005, A001318, A010815, A110654, A235963, A238131, A238132.

A238133 := proc(n)
    add( numtheory[tau](k)*A010815(n-k),k=0..n) ;
end proc: # R. J. Mathar, Jun 18 2016
# second Maple program:
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-> (p-> p[4]-p[3])(b(n$2)):
seq(a(n), n=0..100);  # Alois P. Heinz, Jun 18 2016

Table[SeriesCoefficient[QPochhammer[x] (Log[1 - x] + QPolyGamma[1, x])/Log[x], {x, 0, n}], {n, 0, 80}] (* Vladimir Reshetnikov, Nov 20 2016 *)

a(n) = Sum_{k=0..A235963(n)-1} (-1)^A110654(k) * A000005(n-A001318(k)).
G.f.: Product_{k>=1} (1-x^k) * Sum_{k>=1} x^k/(1-x^k).
G.f.: (x)_inf * (log(1-x) + psi_x(1))/log(x), where psi_q(z) is the q-digamma function, (q)_inf is the q-Pochhammer symbol (the Euler function).

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

Original entry on oeis.org

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

A238450 Triangle read by rows: T(n,k) is the number of k’s in all partitions of n into an odd number of distinct parts.

Original entry on oeis.org

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

Offset: 1

Mircea Merca, Feb 26 2014

			n\k | 1 2 3 4 5 6 7 8 9 10
   1: 1
   2: 0 1
   3: 0 0 1
   4: 0 0 0 1
   5: 0 0 0 0 1
   6: 1 1 1 0 0 1
   7: 1 1 0 1 0 0 1
   8: 2 1 1 1 1 0 0 1
   9: 2 2 2 1 1 1 0 0 1
  10: 3 2 2 1 2 1 1 0 0 1

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns k=1..6 are A238208, A238209, A238210, A238211, A238212, A238213.
Row sums are A238131.
Cf. A015716, A067659, A067661, A238451.

T(n,k) = {my(m=n-k); if(m>0, polcoef(prod(j=1, m, 1+x^j + O(x*x^m))/(1+x^k) + prod(j=1, m, 1-x^j + O(x*x^m))/(1-x^k), m)/2, m==0)} \\ Andrew Howroyd, Apr 29 2020

T(n,k) = Sum_{j=1..round(n/(2*k))} A067661(n-(2*j-1)*k) - Sum_{j=1..floor(n/(2*k))} A067659(n-2*j*k).
G.f. of column k: (1/2)*(q^k/(1+q^k))*(-q;q){inf} + (1/2)*(q^k/(1-q^k))*(q;q){inf}.
T(n,k) = A015716(n,k) - A238451(n,k). - Andrew Howroyd, Apr 29 2020

Terms a(79) and beyond from Andrew Howroyd, Apr 29 2020

A238132 Number of parts in all partitions of n into even number of distinct parts.

Original entry on oeis.org

0, 0, 0, 2, 2, 4, 4, 6, 6, 8, 12, 14, 18, 24, 32, 38, 50, 60, 76, 90, 110, 134, 162, 190, 228, 270, 322, 380, 446, 524, 616, 720, 838, 980, 1134, 1314, 1526, 1760, 2026, 2336, 2676, 3072, 3518, 4020, 4586, 5232, 5948, 6760, 7676, 8698, 9846, 11142, 12578

Offset: 0

Mircea Merca, Feb 18 2014

nonn

			a(8)=6 because the partitions of 8 into even number of distinct parts are: 7+1, 6+2 and 5+3.

Alois P. Heinz, Table of n, a(n) for n = 0..5000
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_e(n).
Eric Weisstein's World of Mathematics, q-Polygamma Function, q-Pochhammer Symbol.

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

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)[3]:
seq(a(n), n=0..60);  # 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))}=A015723(n)-A238131(n).
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

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A238133 Difference between A238131(n) and A238132(n).

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

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A238450 Triangle read by rows: T(n,k) is the number of k’s in all partitions of n into an odd number of distinct parts.

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A238132 Number of parts in all partitions of n into even number of distinct parts.

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