A174712 -id:A174712 - cp's OEIS frontend

A174713 Triangle read by rows, A173305 (A000009 shifted down twice) * A174712 (diagonalized variant of A000041).

1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 2, 4, 2, 2, 3, 5, 3, 4, 3, 6, 4, 4, 3, 5, 8, 5, 6, 6, 5, 10, 6, 8, 6, 5, 7, 12, 8, 10, 9, 10, 7, 15, 10, 12, 12, 10, 7, 11, 18, 12, 16, 15, 15, 14, 11, 22, 15, 20, 18, 20, 14, 11, 15

Offset: 0

Gary W. Adamson, Mar 27 2010

Row sums = A000041, the partition numbers.
The current triangle is the 2nd in an infinite set, followed by A174714 (k=3), and A174715, (k=4); in which row sums of each triangle = A000041.
k-th triangle in the infinite set can be defined as having the sequence:
"Euler transform of ones: (1,1,1,...) interleaved with (k-1) zeros"; shifted down k times (except column 0) in successive columns, then multiplied * triangle A174712, the diagonalized variant of A000041, A174713 begins with A000009 shifted down twice (triangle A173305); where A000009 = the Euler transform of period 2 sequence: [1,0,1,0,...].
Similarly, triangle A174714 begins with A000716 shifted down thrice; where A000716 = the Euler transform of period 3 series: [1,1,0,1,1,0,...]. Then multiply the latter as an infinite lower triangular matrix * A174712, the diagonalized variant of A000041, obtaining triangle A174714 with row sums = A000041.
Case k=4 = triangle A174715 which begins with the Euler transform of period 4 series: [1,1,1,0,1,1,1,0,...], shifted down 4 times in successive columns then multiplied * A174712, the diagonalized variant of A000041.
All triangles in the infinite set have row sums = A000041.
The sequences: "Euler transform of ones interleaved with (k-1) zeros" have the following properties, beginning with k=2:
...
k=2, A000009: = Euler transform of [1,0,1,0,1,0,...] and satisfies
.....A000009. = p(x)/p(x^2), where p(x) = polcoeff A000041; and A000041 =
.....A000009(x) = r(x), then p(x) = r(x) * r(x^2) * r(x^4) * r(x^8) * ...
...
k=3, A000726: = Euler transform of [1,1,0,1,1,0,...] and satisfies
.....A000726(x): = p(x)/p(x^3), and given s(x) = polcoeff A000726, we get
.....A000041(x) = p(x) = s(x) * s(x^3) * s(x^9) * s(x^27) * ...
...
k=4, A001935: = Euler transform of [1,1,1,0,1,1,1,0,...] and satisfies
.....A001935(x) = p(x)/p(x^4) and given t(x) = polcoeff A001935, we get
.....A000041(x) = p(x) = t(x) * t(x^4) * t(x^16) * t(x^64) * ...
...
Also the number of integer partitions of n whose even parts sum to k, for k an even number from zero to n. The version including odd k is A113686. - Gus Wiseman, Oct 23 2023

			First few rows of the triangle =
1;
1;
1, 1;
2, 1;
2, 1, 2;
3, 2, 2;
4, 2, 2, 3;
5, 3, 4, 3;
6, 4, 4, 3, 5;
8, 5, 6, 6, 5;
10, 6, 8, 6, 5, 7;
12, 8, 10, 9, 10, 7;
15, 10, 12, 12, 10, 7, 11;
18, 12, 16, 15, 15, 14, 11;
22, 15, 20, 18, 20, 14, 11, 15;
...
From _Gus Wiseman_, Oct 23 2023: (Start)
Row n = 9 counts the following partitions:
  (9)          (72)        (54)       (63)      (81)
  (711)        (5211)      (522)      (6111)    (621)
  (531)        (3321)      (4311)     (432)     (441)
  (51111)      (321111)    (411111)   (42111)   (4221)
  (333)        (21111111)  (32211)    (3222)    (22221)
  (33111)                  (2211111)  (222111)
  (3111111)
  (111111111)
(End)

Cf. A000009, A173305, A174712, A174714, A174715.
Row sums are A000041.
The odd version is A365067.
The corresponding rank statistic is A366531, odd version A366528.
A000009 counts partitions into odd parts, ranks A066208.
A113685 counts partitions by sum of odd parts, even version A113686.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
Cf. A035363, A066967, A130780, A171966, A182616, A366527, A366533.

Table[Length[Select[IntegerPartitions[n],Total[Select[#,EvenQ]]==k&]],{n,0,15},{k,0,n,2}] (* Gus Wiseman, Oct 23 2023 *)

As infinite lower triangular matrices, A173305 * A174712.

T(n,k) = A000009(n-2k) * A000041(k). - Gus Wiseman, Oct 23 2023

A174714 Triangle read by rows, Q*M; Q = an infinite lower triangular matrix with A000726 shifted down thrice, M = triangle A174712, the diagonalized variant of A000041.

Original entry on oeis.org

1, 1, 2, 2, 1, 4, 1, 5, 2, 7, 2, 2, 9, 4, 2, 13, 5, 4, 16, 7, 4, 3, 22, 9, 8, 3, 27, 13, 10, 6, 36, 16, 14, 6, 5, 44, 22, 18, 12, 5, 57, 27, 26, 15, 10

Offset: 0

Gary W. Adamson, Mar 27 2010

Refer to comments in A174713.

Row sums = A000041, the partition numbers.

			First few rows of the triangle =
1;
1;
2;
2, 1;
4, 1;
5, 2;
7, 2, 2;
9, 4, 2;
13, 5, 4;
16, 7, 4, 3;
22, 9, 8, 3;
27, 13, 10, 6;
36, 16, 14, 6, 5;
44, 22, 18, 12, 5;
57, 27, 26, 15, 10;
...

Cf. A000041, A000726, A174712, A174713, A174715.

Let Q = an infinite lower triangular matrix with A000726, (Euler transform of [1,1,0,1,1,0,...]) in each column shifted down thrice from the (k-1)-th column, excepting column 0. Let M = triangle A174712, the diagonalized variant of A000041. Then triangle A174714 = Q*M.

A174715 Triangle read by rows, Q*M as infinite lower triangular matrices. Q = A001935 shifted down four times by columns, M = A174712.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 6, 1, 9, 2, 12, 3, 16, 4, 2, 22, 6, 2, 29, 9, 4, 38, 12, 6, 50, 16, 8, 3, 64, 22, 12, 3, 82, 29, 18, 6

Offset: 0

Gary W. Adamson, Mar 27 2010

Row sums = A000041, the partition numbers.

Refer to comments in A174713.

			First few rows of the triangle =
  1;
  1;
  2;
  3;
  4, 1;
  6, 1;
  9, 2;
  12, 3;
  16, 4, 2;
  22, 6, 2;
  29, 9, 4;
  38, 12, 6;
  50, 16, 8, 3;
  64, 22, 12, 3;
  82, 29, 18, 6;
  ...

Cf. A174712, A174712, A174713, A174714

As an irregular triangle generated from a product Q*M of infinite lower triangular matrices: Q = A001935 in every column (except column 0), shifted down four times from the previous column. M = A174712, the diagonalized variant of A000041 (the partition numbers as the right border, and the rest zeros.)

A174739 Triangle read by rows, a partition number generator; A145006 * the diagonalized variant of A000041, (A174712).

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 0, 0, 2, 3, -1, 0, 0, 3, 5, 0, -1, 0, 0, 5, 7, -1, 0, -2, 0, 0, 7, 11, 0, -1, 0, -3, 0, 0, 11, 15, 0, 0, -2, 0, -5, 0, 15, 22, 0, 0, 0, -3, 0, -7, 0, 0, 22, 30, 0, 0, 0, 0, -5, 0, -11, 0, 0, 30, 42, 1, 0, 0, 0, 0, -7, 0, -15, 0, 0, 42, 56, 0, 1, 0, 0, 0, 0, -11, 0, -22, 0, 0, 0, 0

Offset: 1

Gary W. Adamson, Mar 28 2010

Row sums = the partition numbers, A000041 starting with offset 1.
The triangle demonstrates an equivalency to Euler's pentagonal recurrence
relation, such that sum of n-th row terms = rightmost term of next row, a
partition number.
Contribution from Gary W. Adamson, Mar 28 2010: (Start)
A174739 is equivalent to Euler's pentagonal theorem in triangular form.
For example, row 9 = (0, 0, -2, 0, -5, 0, 0, 15, 22) or: p(9) = 30 = p(8)
+ p(7) - p(4) - p(2). (End)

			First few rows of the triangle =
1;
1, 1;
0, 1, 2;
0, 0, 2, 3;
-1, 0, 0, 3, 5;
0, -1, 0, 0, 5, 7;
-1, 0, -2, 0, 0, 7, 11;
0, -1, 0, -3, 0, 0, 11, 15;
0, 0, -2, 0, -5, 0, 0, 15, 22;
0, 0, 0, -3, 0, -7, 0, 0, 22, 30;
0, 0, 0, 0, -5, 0, -11, 0, 0, 30, 42;
1, 0, 0, 0, 0, -7, 0, -15, 0, 0, 42, 56;
0, 1, 0, 0, 0, 0, -11, 0, 22, 0, 0, 56, 77;
0, 0, 2, 0, 0, 0, 0, -15, 0, -30, 0, 0, 77, 101;
1, 0, 0, 3, 0, 0, 0, 0, -22, 0, -42, 0, 0, 101, 135;
0, 1, 0, 0, 5, 0, 0, 0, 0, -30, 0, -56, 0, 0, 135, 176;
0, 0, 2, 0, 0, 7, 0, 0, 0, 0, -42, 0, -77, 0, 0, 176, 231;
0, 0, 0, 3, 0, 0, 11, 0, 0, 0, 0, -56, 0, -101, 0, 0, 231, 297;
...

Cf. A000041, A145006, A174712

Given triangle A145006, delete the first "1", = triangle Q. With M = A174712,
the diagonalize variant of the partition numbers, perform Q*M as infinite lower
triangular matrices.

A058694 Partial products p(0)p(1)...*p(n) of partition numbers A000041.

Original entry on oeis.org

1, 1, 2, 6, 30, 210, 2310, 34650, 762300, 22869000, 960498000, 53787888000, 4141667376000, 418308404976000, 56471634671760000, 9939007702229760000, 2295910779215074560000, 681885501426877144320000, 262525918049347700563200000, 128637699844180373275968000000

Offset: 0

N. J. A. Sloane, Dec 30 2000

nonn

a(n) gives the number of partitions P(V(n)) of V(n)=[1,2,3,...,n]. A partition P(V(n)) acts on the components of V(n), i.e., the components of V(n) are partitioned. Therefore a(n) results as the product of the number of partitions P(i) of the component v(i)=i with i=1,...,n. For example, a(3) = 6 because we have 6 list partitions for the list V(n=3)=[1,2,3]: [[1], [1, 1], [2, 1]], [[1], [1, 1], [1, 1, 1]], [[1], [1, 1], [3]], [[1], [2], [2, 1]], [[1], [2], [1, 1, 1]], [[1], [2], [3]]. - Thomas Wieder, Sep 29 2007

Equals the eigensequence of triangle A174712; i.e., Triangle A174712 * A058694 preceded by a 1 shifts left. - Gary W. Adamson, Mar 27 2010

Alois P. Heinz, Table of n, a(n) for n = 0..150
Vaclav Kotesovec, The partition factorial constant and asymptotics of the sequence A058694
Eric Weisstein's MathWorld, Hurwitz Zeta Function

Cf. A000041, A000070, A133018, A259314, A259373, A174712.

a:= proc(n) option remember;
       combinat[numbpart](n)*`if`(n>0, a(n-1), 1)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 21 2012
#
# The constant S in the Maple notation
evalf(Zeta(0, -1/2, 23/24)*sqrt(2/3)*Pi - Zeta(0, 1/2, 23/24)*sqrt(3/2)/Pi+3*(D(GAMMA))(23/24)/(4*Pi^2*GAMMA(23/24)) - (Sum(Zeta(0, j/2, 23/24)*(sqrt(3/2)/Pi)^j/j, j=3..infinity)), 60); # Vaclav Kotesovec, Jun 24 2015

Table[Product[PartitionsP[k], {k, 1, n}], {n, 1, 33}] (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)

a(n)=prod(k=2,n, numbpart(k)) \\ Charles R Greathouse IV, Jan 14 2017

a(n) ~ C * Product_{k=1..n} (exp(Pi*sqrt(2/3*(k-1/24))) / (4*sqrt(3)*(k-1/24)) * (1 - sqrt(3/(2*(k-1/24)))/Pi)), where C = 0.9110167313322499518... is the partition factorial constant A259314. - Vaclav Kotesovec, Jun 24 2015

a(n) ~ C * Gamma(23/24) / (n^(n + 11/24 + 3/(4*Pi^2)) * 2^(2*n) * 3^(n/2) * sqrt(2*Pi)) * exp(Pi*(2*n/3)^(3/2) + n + (11*Pi/(12*sqrt(6)) - sqrt(6)/Pi)*sqrt(n) + S), where C = A259314 and S = Zeta(-1/2, 23/24)*sqrt(2/3)*Pi - Zeta(1/2, 23/24)*sqrt(3/2)/Pi + 3*Gamma'(23/24)/(4*Pi^2*Gamma(23/24)) - Sum_{j>=3} Zeta(j/2, 23/24)*(sqrt(3/2)/Pi)^j/j = -0.02541933397793652709903012019225640813047573968579474..., Zeta is the Hurwitz Zeta Function, in Maple notation Zeta(0,z,v), in Mathematica notation Zeta[z,v], equivalently HurwitzZeta[z,v]. - Vaclav Kotesovec, Jun 24 2015

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A174713 Triangle read by rows, A173305 (A000009 shifted down twice) * A174712 (diagonalized variant of A000041).

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A174714 Triangle read by rows, Q*M; Q = an infinite lower triangular matrix with A000726 shifted down thrice, M = triangle A174712, the diagonalized variant of A000041.

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A174715 Triangle read by rows, Q*M as infinite lower triangular matrices. Q = A001935 shifted down four times by columns, M = A174712.

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A174739 Triangle read by rows, a partition number generator; A145006 * the diagonalized variant of A000041, (A174712).

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A058694 Partial products p(0)*p(1)*...*p(n) of partition numbers A000041.

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A058694 Partial products p(0)p(1)...*p(n) of partition numbers A000041.