A235963 -id:A235963 - cp's OEIS frontend

A175003 Triangle read by rows demonstrating Euler's pentagonal theorem for partition numbers.

1, 1, 1, 2, 1, 3, 2, 5, 3, -1, 7, 5, -1, 11, 7, -2, -1, 15, 11, -3, -1, 22, 15, -5, -2, 30, 22, -7, -3, 42, 30, -11, -5, 56, 42, -15, -7, 1, 77, 56, -22, -11, 1, 101, 77, -30, -15, 2, 135, 101, -42, -22, 3, 1, 176, 135, -56, -30, 5, 1, 231, 176, -77, -42, 7, 2

Offset: 1

Gary W. Adamson, Apr 03 2010

Row sums = A000041 starting with offset 1.
Sum of n-th row terms = leftmost term of next row, such that terms in each row demonstrate Euler's pentagonal theorem.
Let Q = triangle A027293 with partition numbers in each column.
Let M = a diagonalized variant of A080995 as the characteristic function of the generalized pentagonal numbers starting with offset 1: (1, 1, 0, 0, 1,...)
Sign the 1's: (++--++...) getting (1, 1, 0, 0, -1, 0, -1,...) which is the diagonal of matrix M, (as an infinite lower triangular matrix with the rest zeros).
Triangle A175003 = Q*M, with deleted zeros.
Column k starts at row A001318(k). - Omar E. Pol, Sep 21 2011
From Omar E. Pol, Apr 22 2014: (Start)
Row n has length A235963(n).
For Euler's pentagonal theorem for the sum of divisors see A238442.
Note that both of Euler's pentagonal theorems refer to generalized pentagonal numbers (A001318), not to pentagonal numbers (A000326). (End)

			Triangle begins:
    1;
    1,   1;
    2,   1;
    3,   2;
    5,   3,  -1;
    7,   5,  -1;
   11,   7,  -2,  -1;
   15,  11,  -3,  -1;
   22,  15,  -5,  -2;
   30,  22,  -7,  -3;
   42,  30, -11,  -5;
   56,  42, -15,  -7,   1;
   77,  56, -22, -11,   1;
  101,  77, -30, -15,   2;
  ...

Cf. A000041, A080995, A027293, A238442.

T(n,k) = A057077(k-1)*A000041(A195310(n,k)), n >= 1, k >= 1. - Omar E. Pol, Sep 21 2011

Corrected and extended by Omar E. Pol, Feb 14 2013

A238442 Triangle read by rows demonstrating Euler's pentagonal theorem for the sum of divisors.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 3, 7, 4, -5, 6, 7, -1, 12, 6, -3, -7, 8, 12, -4, -1, 15, 8, -7, -3, 13, 15, -6, -4, 18, 13, -12, -7, 12, 18, -8, -6, 12, 28, 12, -15, -12, 1, 14, 28, -13, -8, 3, 24, 14, -18, -15, 4, 15, 24, 24, -12, -13, 7, 1, 31, 24, -28, -18, 6, 3

Offset: 1

Omar E. Pol, Feb 26 2014

The law found by Leonhard Euler for the sum of divisors of n is that S(n) = S(n - 1) + S(n - 2) - S(n - 5) - S(n - 7) + S(n - 12) + S(n - 15) - S(n - 22) - S(n - 26) + S(n - 35) + S(n - 40) + ..., where the constants are the positive generalized pentagonal numbers, and S(0) = n, which is also a positive member of A001318.
Therefore column k lists A001318(k) together with the elements of A000203, starting at row A001318(k), but with all elements of column k multiplied by A057077(k-1).
The first element of column k is A057077(k-1)*A001318(k)which is also the last term of row A001318(k).
For Euler's pentagonal theorem for the partition numbers see A175003.
Note that both of Euler's pentagonal theorems refer to generalized pentagonal numbers (A001318), not to pentagonal numbers (A000326).

			Triangle begins:
   1;
   1,   2;
   3,   1;
   4,   3;
   7,   4,  -5;
   6,   7,  -1;
  12,   6,  -3,  -7;
   8,  12,  -4,  -1;
  15,   8,  -7,  -3;
  13,  15,  -6,  -4;
  18,  13, -12,  -7;
  12,  18,  -8,  -6,  12;
  28,  12, -15, -12,   1;
  14,  28, -13,  -8,   3;
  24,  14, -18, -15,   4,  15;
  24,  24, -12, -13,   7,   1;
  31,  24, -28, -18,   6,   3;
  18,  31, -14, -12,  12,   4;
  39,  18, -24, -28,   8,   7;
  20,  39, -24, -14,  15,   6;
  42,  20, -31, -24,  13,  12;
  32,  42, -18, -24,  18,   8, -22;
  36,  32, -39, -31,  12,  15,  -1;
  24,  36, -20, -18,  28,  13,  -3;
  60,  24, -42, -39,  14,  18,  -4;
  31,  60, -32, -20,  24,  12,  -7, -26;
  ...
For n = 21 the sum of divisors of 21 is 1 + 3 + 7 + 21 = 32. On the other hand, from Euler's Pentagonal Number Theorem we have that the sum of divisors of 21 is S_21 = S_20 + S_19 - S_16 - S_14 + S_9 + S_6, the same as the sum of the 21st row of triangle: 42 + 20 - 31 - 24 + 13 + 12 = 32, equaling the sum of divisors of 21.
For n = 22 the sum of divisors of 22 is 1 + 2 + 11 + 22 = 36. On the other hand, from Euler's Pentagonal Number Theorem we have that the sum of divisors of 22 is S_22 = S_21 + S_20 - S_17 - S_15 + S_10 + S_7 - S_0, the same as the sum of the 22nd row of triangle is 32 + 42 - 18 - 24 + 18 + 8 - 22 = 36, equaling the sum of divisors of 22. Note that S_0 = n, hence in this case S_0 = 22.

L. Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009, p. 8.
L. Euler, De mirabilibus proprietatibus numerorum pentagonalium
L. Euler, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs
L. Euler, Discovery of a most extraordinary law of numbers, relating to the sum of their divisors
L. Euler, Observatio de summis divisorum, p. 8.
L. Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
L. Euler, J. Bell, A demonstration of a theorem on the order observed in the sums of divisors, arXiv:math/0507201 [math.HO], 2005-2009.
Index entries for sequences related to sigma(n)

Row sums give A000203, the sum of divisors of n.
Row n has length A235963(n).
Cf. A001318, A027750, A057077, A175003, A196020, A237270, A237273.

rows = m = 18;
a057077[n_] := {1, 1, -1, -1}[[Mod[n, 4] + 1]];
a001318[n_] := (1/8)((2n + 1) Mod[n, 2] + 3n^2 + 2n);
a235963[n_] := Flatten[Table[k, {k, 0, m}, {(k+1)/(Mod[k, 2]+1)}]][[n+1]];
T[n_, k_] := If[n == a001318[k] && k == a235963[n], a001318[k] a057077[k - 1], a057077[k - 1] DivisorSigma[1, n - a001318[k]]];
Table[T[n, k], {n, 1, m}, {k, 1, a235963[n]}] // Flatten (* Jean-François Alcover, Nov 29 2018 *)

T(n,k) = A057077(k-1)*A001318(k), if n = A001318(k) and k = A235963(n). Otherwise T(n,k) = A057077(k-1)*A000203(n - A001318(k)), n >= 1, 1 <= k <= A235963(n).

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

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

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

Original entry on oeis.org

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).

cp's OEIS Frontend

A175003 Triangle read by rows demonstrating Euler's pentagonal theorem for partition numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A238442 Triangle read by rows demonstrating Euler's pentagonal theorem for the sum of divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula