A008284 -id:A008284 - cp's OEIS frontend

A238312 Square row sums of the table A072233 (A008284).

1, 1, 2, 3, 7, 11, 25, 41, 82, 142, 260, 436, 785, 1287, 2199, 3592, 5959, 9511, 15453, 24268, 38565, 59838, 93232, 142589, 219089, 330848, 500658, 748140, 1117856, 1651987, 2441484, 3572470, 5223653, 7576447, 10971112, 15775735, 22649645, 32307553, 46001087, 65138447, 92045412

Offset: 0

Emanuele Munarini, Feb 24 2014

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 81 terms from Vincenzo Librandi)

Cf. A000290, A008284, A072233.

pnkList[n_] := Table[Length[IntegerPartitions[n, {k}]], {k, 0, n}]
Table[Total[Map[#^2 &, pnkList[n]]], {n, 0, 40}]

a(n) = Sum_{k=0..n} p(n,k)^2, where p(n,k) is the number of partitions of n into k positive parts (A072233, A008284).

A325514 Heinz number of row n of the triangle of partition numbers A008284.

Original entry on oeis.org

2, 2, 4, 8, 24, 72, 600, 4200, 101640, 2042040, 107869080, 6435365640, 644779672680, 62219208188280, 14408598135902520, 3195700205016233640, 1246437353286578234760, 527744165981695537415640, 417665868515500206974318760, 314096677106179199154141208440

Offset: 0

Gus Wiseman, May 07 2019

nonn

The Heinz number of a positive integer sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
             2: {1}
             2: {1}
             4: {1,1}
             8: {1,1,1}
            24: {1,1,1,2}
            72: {1,1,1,2,2}
           600: {1,1,1,2,3,3}
          4200: {1,1,1,2,3,3,4}
        101640: {1,1,1,2,3,4,5,5}
       2042040: {1,1,1,2,3,4,5,6,7}
     107869080: {1,1,1,2,3,5,5,7,8,9}
    6435365640: {1,1,1,2,3,5,5,7,10,10,11}
  644779672680: {1,1,1,2,3,5,6,7,11,12,13,15}

Index entries for sequences related to Heinz numbers

Cf. A000041, A002569, A008284, A056239, A112798, A145519, A215366, A325500, A325502, A325503, A325512, A325514.

Times@@@Table[If[n>0&&k==0,1,Prime[Length[IntegerPartitions[n,{k}]]]],{n,0,20},{k,0,n}]

A001221(a(n)) = A325512(n).
A061395(a(n)) = A002569(n).
A056239(a(n)) = A000041(n).

A327028 T(n, k) = k! * Sum_{d|n} phi(d) * A008284(n/d, k) for n >= 1, T(0, k) = 0^k. Triangle read by rows for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 2, 6, 0, 4, 6, 6, 24, 0, 5, 4, 12, 24, 120, 0, 6, 12, 24, 48, 120, 720, 0, 7, 6, 24, 72, 240, 720, 5040, 0, 8, 16, 36, 144, 360, 1440, 5040, 40320, 0, 9, 12, 54, 144, 600, 2160, 10080, 40320, 362880

Offset: 0

Peter Luschny, Aug 20 2019

			[0] 1
[1] 0, 1
[2] 0, 2,  2
[3] 0, 3,  2,  6
[4] 0, 4,  6,  6,  24
[5] 0, 5,  4, 12,  24, 120
[6] 0, 6, 12, 24,  48, 120,  720
[7] 0, 7,  6, 24,  72, 240,  720,  5040
[8] 0, 8, 16, 36, 144, 360, 1440,  5040, 40320
[9] 0, 9, 12, 54, 144, 600, 2160, 10080, 40320, 362880

Cf. A008284, A318144, A000142 (main diagonal), A327025 (row sums), A327029.

A327028 := (n,k) -> `if`(n=0, 1, k!*add(phi(d)*A008284(n/d, k), d = divisors(n))):
seq(seq(A327028(n, k), k=0..n), n=0..9);

A327028[0 , k_] := 1;
A327028[n_, k_] := DivisorSum[n, EulerPhi[#] A318144[n/#, k] (-1)^k &];
Table[A327028[n, k], {n, 0,  9}, {k, 0,  n}] // Flatten

# uses[DivisorTriangle from A327029]
from sage.combinat.partition import number_of_partitions_length
def A318144Abs(n, k): return number_of_partitions_length(n, k)*factorial(k)
DivisorTriangle(euler_phi, A318144Abs, 10)

A039808 Shifts left under transform T where Ta is product of Partition Triangle A008284 with a.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 47, 109, 258, 609, 1447, 3434, 8175, 19449, 46327, 110330, 262882, 626317, 1492515, 3556540, 8475679, 20198343, 48136276, 114716767, 273393363, 651550405, 1552783297, 3700609736, 8819356189

Offset: 1

Christian G. Bower, Feb 15 1999

Cf. A008284, A039809.

G.f.: x + x * Sum_{n>=1} a(n) * x^n / Product_{j=1..n} (1 - x^j). - Ilya Gutkovskiy, Jul 22 2021

A050302 Matrix 8th power of partition triangle A008284.

Original entry on oeis.org

1, 8, 1, 36, 8, 1, 148, 44, 8, 1, 498, 184, 44, 8, 1, 1590, 682, 192, 44, 8, 1, 4586, 2236, 718, 192, 44, 8, 1, 12644, 6822, 2420, 726, 192, 44, 8, 1, 32775, 19346, 7476, 2456, 726, 192, 44, 8, 1, 81901, 52177, 21646, 7660, 2464, 726, 192, 44, 8, 1, 196085

Offset: 1

Christian G. Bower, Aug 15 1999

			1; 8,1; 36,8,1; 148,44,8,1; ...

Cf. A038497, A038498, A039805-A039807. A050301-A050304. a(n, 1) = A024208(n) (first column).

A055889 CIK transform of partition triangle A008284.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 4, 4, 5, 1, 4, 8, 8, 7, 1, 6, 13, 19, 16, 13, 1, 6, 18, 32, 40, 32, 19, 1, 8, 24, 56, 80, 90, 64, 35, 1, 8, 33, 80, 148, 196, 192, 128, 59, 1, 10, 40, 120, 247, 396, 464, 420, 256, 107, 1, 10, 50, 162, 392, 712, 1008, 1088, 896, 512, 187, 1, 12, 61

Offset: 1

Christian G. Bower, Jun 09 2000

			1; 1,2; 1,2,3; 1,4,4,5; 1,4,8,8,7; ...

C. G. Bower, Transforms (2)

Row sums give A055890.

A181845 Triangle read by rows: T(n,k) = max_{c in P(n,n-k+1)} lcm(c) where P(n,m) = A008284(n,m) is the number of partitions of n into m parts.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 6, 5, 1, 2, 3, 6, 5, 6, 1, 2, 3, 6, 6, 12, 7, 1, 2, 3, 6, 6, 12, 15, 8, 1, 2, 3, 6, 6, 12, 15, 20, 9, 1, 2, 3, 6, 6, 12, 15, 30, 21, 10, 1, 2, 3, 6, 6, 12, 15, 30, 21, 30, 11, 1, 2, 3, 6, 6, 12, 15, 30, 30, 60, 35, 12

Offset: 1

Peter Luschny, Dec 07 2010

See A181842 for the definition of 'partition'. T(n,k) is also the triangle read by rows: T(n,k) = max_{c in C(n,n-k+1)} lcm(c) where C(n,m) is the set of all m-tuples of positive integers whose elements sum to n where the C(n,k) = A007318(n-1,k-1) are called compositions of n of size k.

			[1]   1
[2]   1   2
[3]   1   2   3
[4]   1   2   3   4
[5]   1   2   3   6   5
[6]   1   2   3   6   5   6
[7]   1   2   3   6   6   12   7
[8]   1   2   3   6   6   12   15   8
[9]   1   2   3   6   6   12   15   20   9

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Cf. A181842, A181843, A181844.

with(combstruct):
a181845_row := proc(n) local k,L,l,R,part;
R := NULL;
for k from 1 to n do
   L := 0;
   part := iterstructs(Partition(n),size=n-k+1):
# alternatively (but slower)
# part := iterstructs(Composition(n), size=n-k+1):
   while not finished(part) do
      l := nextstruct(part);
      L := max(L,ilcm(op(l)));
   od;
   R := R,L;
od;
R end:

Row(n)={my(v=vector(n)); forpart(p=n, my(i=#p); v[i]=max(v[i], lcm(Vec(p)))); Vecrev(v)}
{ for(n=1, 10, print(Row(n))) } \\ Andrew Howroyd, Apr 20 2021

Terms a(56) and beyond from Andrew Howroyd, Apr 20 2021

A238313 Alternating square row sums of the table A072233 (A008284).

Original entry on oeis.org

1, 1, 0, 1, 3, 1, 3, 3, 10, 18, 12, 26, 39, 57, 59, 116, 201, 219, 325, 416, 625, 810, 1074, 1447, 2345, 3078, 3530, 5084, 6790, 9063, 11674, 15580, 20887, 27537, 33640, 45065, 61297, 76883, 96889, 126243, 169268, 210005, 262068, 337445, 438197, 552346, 686794, 865904, 1128611, 1407533, 1732572

Offset: 0

Emanuele Munarini, Feb 24 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..90

Cf. A072233, A008284.

pnkList[n_] := Table[Length[IntegerPartitions[n, {k}]], {k,0,n}]
Table[Total[Table[(-1)^(n-k),{k,0,n}] Map[#^2 &, pnkList[n]]], {n,0,50}]

a(n) = sum((-1)^(n-k)*p(n,k)^2,k=0..n), where p(n,k) is the number of partitions of n into k positive parts (A072233, A008284).

A238314 Binomial transform of the squared rows of the table A072233 (A008284).

Original entry on oeis.org

1, 1, 3, 7, 33, 91, 388, 1163, 4231, 13297, 44694, 136621, 444535, 1335335, 4149785, 12327698, 37154245, 108185961, 318923590, 913506701, 2633793550, 7443298426, 21073435606, 58715695683, 163805615535, 450730653566, 1239947467778, 3374934052348

Offset: 0

Emanuele Munarini, Feb 24 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..80

Cf. A072233, A008284.

pnkList[n_] := Table[Length[IntegerPartitions[n, {k}]], {k, 0, n}]
Table[Total[Table[Binomial[n,k],{k,0,n}] Map[#^2 &, pnkList[n]]], {n,0,40}]

a(n) = sum(binomial(n,k)*p(n,k)^2,k=0..n), where p(n,k) is the number of partitions of n into k positive parts (A072233, A008284).

A327025 a(n) = Sum_{k=0..n} k! * Sum_{d|n} phi(d) * A008284(n/d, k).

Original entry on oeis.org

1, 1, 4, 11, 40, 165, 930, 6109, 47364, 416259, 4092332, 44424105, 527512456, 6798907261, 94504292830, 1408973416983, 22426222792568, 379522092608193, 6804315178122972, 128828842646944153, 2568533750232696472, 53788282243854348683, 1180357162840669081094

Offset: 0

Peter Luschny, Aug 23 2019

nonn

Row sums of A327028.

cp's OEIS Frontend

A238312 Square row sums of the table A072233 (A008284).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A325514 Heinz number of row n of the triangle of partition numbers A008284.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A327028 T(n, k) = k! * Sum_{d|n} phi(d) * A008284(n/d, k) for n >= 1, T(0, k) = 0^k. Triangle read by rows for 0 <= k <= n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

SageMath

A039808 Shifts left under transform T where Ta is product of Partition Triangle A008284 with a.

Views

Author

Keywords

Crossrefs

Formula

A050302 Matrix 8th power of partition triangle A008284.

Views

Author

Keywords

Examples

Crossrefs

A055889 CIK transform of partition triangle A008284.

Views

Author

Keywords

Examples

Links

Crossrefs

A181845 Triangle read by rows: T(n,k) = max_{c in P(n,n-k+1)} lcm(c) where P(n,m) = A008284(n,m) is the number of partitions of n into m parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Extensions

A238313 Alternating square row sums of the table A072233 (A008284).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A238314 Binomial transform of the squared rows of the table A072233 (A008284).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula