A323300 -id:A323300 - cp's OEIS frontend

A101509 Binomial transform of tau(n) (see A000005).

1, 3, 7, 16, 35, 75, 159, 334, 696, 1442, 2976, 6123, 12562, 25706, 52492, 107014, 217877, 443061, 899957, 1826078, 3701783, 7498261, 15178255, 30706320, 62085915, 125465715, 253415981, 511608490, 1032427637, 2082680887, 4199956101, 8467124805, 17064784905, 34382825363, 69256687719, 139465867773

Offset: 0

Paul Barry, Dec 05 2004

Row sums of A101508.

Also: Number of matrices with positive integer coefficients such that the sum of all entries equals n+1, cf. link "Partitions and A101509". - M. F. Hasler, Jan 14 2009

			From _Gus Wiseman_, Jan 16 2019: (Start)
The a(3) = 16 ways to arrange the parts of an integer partition of 4 into a matrix:
  [4] [1 3] [3 1] [2 2] [1 1 2] [1 2 1] [2 1 1] [1 1 1 1]
.
  [1] [3] [2] [1 1]
  [3] [1] [2] [1 1]
.
  [1] [1] [2]
  [1] [2] [1]
  [2] [1] [1]
.
  [1]
  [1]
  [1]
  [1]
(End)

M. F. Hasler, Table of n, a(n) for n = 0..500
L. Manor, M. F. Hasler, Partitions and A101509. SeqFan list, Jan 14 2009
N. J. A. Sloane, Transforms

Cf. A000005 (tau), A101508, A160399.

Cf. A000219, A047966, A053529, A319066, A323300, A323301, A323307, A323429.

bintr:= proc(p) proc(n) add(p(k) *binomial(n, k), k=0..n) end end:
a:= bintr(n-> numtheory[tau](n+1)):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 30 2011

a[n_] := Sum[DivisorSigma[0, k+1]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 18 2017 *)

A101509(n) = sum( k=0,n, numdiv(k+1)*binomial(n,k)) \\ M. F. Hasler, Jan 14 2009

a(n) = Sum_{k=0..n, Sum_{i=0..n, if(mod(i+1, k+1)=0, binomial(n, i), 0)}}.
G.f.: 1/x * Sum_{n>=1} z^n/(1-z^n) (Lambert series) where z=x/(1-x). - Joerg Arndt, Jan 30 2011
a(n) ~ 2^n * (log(n/2) + 2*gamma), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 07 2020

A323433 Number of ways to split an integer partition of n into consecutive subsequences of equal length.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 25, 34, 54, 74, 109, 146, 211, 276, 381, 501, 675, 871, 1156, 1477, 1926, 2447, 3142, 3957, 5038, 6291, 7918, 9839, 12277, 15148, 18773, 23027, 28333, 34587, 42284, 51357, 62466, 75503, 91344, 109971, 132421, 158755, 190365, 227354, 271511

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

			The a(5) = 14 split partitions:
  [5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]
.
  [4] [3] [2 1]
  [1] [2] [1 1]
.
  [3] [2]
  [1] [2]
  [1] [1]
.
  [2]
  [1]
  [1]
  [1]
.
  [1]
  [1]
  [1]
  [1]
  [1]

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A000005, A000219, A008284, A101509, A316245, A319066, A323295, A323300, A323307, A323429, A323434.

b:= proc(n, i, t) option remember; `if`(n=0 or i=1, numtheory
      [tau](t+n), b(n, i-1, t)+b(n-i, min(n-i, i), t+1))
    end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 15 2019

Table[Sum[Length[Divisors[Length[ptn]]],{ptn,IntegerPartitions[n]}],{n,30}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1,
     DivisorSigma[0, t+n], b[n, i-1, t] + b[n-i, Min[n-i, i], t+1]];
a[n_] := If[n == 0, 1, b[n, n, 0]];
a /@ Range[0, 50] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

my(N=66, x='x+O('x^N)); Vec(1+sum(k=1, N, numdiv(k)*x^k/prod(j=1, k, 1-x^j))) \\ Seiichi Manyama, Jan 21 2022

my(N=66, x='x+O('x^N)); Vec(1+sum(i=1, N, sum(j=1, N\i, x^(i*j)/prod(k=1, i*j, 1-x^k)))) \\ Seiichi Manyama, Jan 21 2022

a(n) = Sum_y A000005(k), where the sum is over all integer partitions of n and k is the number of parts.
From Seiichi Manyama, Jan 21 2022: (Start)
G.f.: 1 + Sum_{k>=1} A000005(k) * x^k/Product_{j=1..k} (1-x^j).
G.f.: 1 + Sum_{i>=1} Sum_{j>=1} x^(i*j)/Product_{k=1..i*j} (1-x^k). (End)
a(n) = Sum_{i=1..n} Sum_{j=1..n} A008284(n,i*j). - Ridouane Oudra, Apr 13 2023

A323436 Number of plane partitions whose parts are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 5, 1, 4, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 4, 1, 7, 2, 2, 2, 8, 1, 2, 2, 5, 1, 4, 1, 3, 3, 2, 1, 7, 2, 4, 2, 3, 1, 7, 2, 5, 2, 2, 1, 8, 1, 2, 3, 11, 2, 4, 1, 3, 2, 4, 1, 12, 1, 2, 4, 3, 2, 4, 1, 7, 5, 2, 1, 8, 2, 2

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are weakly decreasing.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(120) = 12 plane partitions:
  32111
.
  311   321   3111   3211
  21    11    2      1
.
  31   32   311   321
  21   11   2     1
  1    1    1     1
.
  31   32
  2    1
  1    1
  1    1
.
  3
  2
  1
  1
  1

Cf. A000085, A000219, A003293, A056239, A112798, A114736, A117433, A138178, A296188, A299968.

Cf. A323300, A323429, A323437, A323438, A323439.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Length[Select[ptnplane[y],And[And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,100}]

A323351 Number of ways to fill a (not necessarily square) matrix with n zeros and ones.

Original entry on oeis.org

1, 2, 8, 16, 48, 64, 256, 256, 1024, 1536, 4096, 4096, 24576, 16384, 65536, 131072, 327680, 262144, 1572864, 1048576, 6291456, 8388608, 16777216, 16777216, 134217728, 100663296, 268435456, 536870912, 1610612736, 1073741824, 8589934592, 4294967296, 25769803776

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

			The a(3) = 16 matrices:
  [000] [001] [010] [011] [100] [101] [110] [111]
.
  [0] [0] [0] [0] [1] [1] [1] [1]
  [0] [0] [1] [1] [0] [0] [1] [1]
  [0] [1] [0] [1] [0] [1] [0] [1]

Cf. A000005, A049311, A120733, A323295, A323300.

Mathematica
```
Table[2^n*DivisorSigma[0,n],{n,10}]
```

a(n) = if (n==0, 1, 2^n*numdiv(n)); \\ Michel Marcus, Jan 15 2019

a(n) = 2^n * A000005(n) for n > 0, a(0) = 1.

G.f.: 1 + Sum_{k>=1} 2^k*x^k/(1 - 2^k*x^k). - Ilya Gutkovskiy, May 23 2019

A323438 Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are weakly increasing.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 3, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 4, 1, 7, 2, 2, 2, 8, 1, 2, 2, 7, 1, 4, 1, 4, 4, 2, 1, 12, 2, 3, 2, 4, 1, 5, 2, 7, 2, 2, 1, 10, 1, 2, 4, 11, 2, 4, 1, 4, 2, 4, 1, 13, 1, 2, 3, 4, 2, 4, 1, 12, 5, 2, 1, 10, 2

Offset: 1

Gus Wiseman, Jan 16 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(96) = 19 tableaux:
  111112
.
  111   1111   1112   11111   11112
  112   12     11     2       1
.
  11   111   111   112   1111   1112
  11   11    12    11    1      1
  12   2     1     1     2      1
.
  11   11   111   112
  11   12   1     1
  1    1    1     1
  2    1    2     1
.
  11   12
  1    1
  1    1
  1    1
  2    1
.
  1
  1
  1
  1
  1
  2

Wikipedia, Young tableau

Cf. A000085, A000219, A003293, A056239, A112798, A114736, A117433, A138178, A299968, A323300, A323430, A323436, A323438, A323450.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Length[Select[ptnplane[y],And[And@@LessEqual@@@#,And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])]&]],{y,100}]

Sum_{A056239(n) = k} a(k) = A323450(n).

A323295 Number of ways to fill a matrix with the first n positive integers.

Original entry on oeis.org

1, 1, 4, 12, 72, 240, 2880, 10080, 161280, 1088640, 14515200, 79833600, 2874009600, 12454041600, 348713164800, 5230697472000, 104613949440000, 711374856192000, 38414242234368000, 243290200817664000, 14597412049059840000, 204363768686837760000

Offset: 0

Gus Wiseman, Jan 12 2019

nonn

			The a(4) = 72 matrices consist of:
  24 row/column permutations of [1 2 3 4]
+
  4 row/column permutations of [1 2]
                               [3 4]
+
  4 row/column permutations of [1 2]
                               [4 3]
+
  4 row/column permutations of [1 3]
                               [2 4]
+
  4 row/column permutations of [1 3]
                               [4 2]
+
  4 row/column permutations of [1 4]
                               [2 3]
+
  4 row/column permutations of [1 4]
                               [3 2]
+
  24 row/column permutations of [1]
                                [2]
                                [3]
                                [4]

Cf. A000005, A000142, A038041, A053529, A057625, A061095, A120733, A121860.

Cf. A323300, A323301, A323307, A323351.

Join[{1}, Table[DivisorSigma[0, n]*n!, {n, 30}]]

a(n) = if (n==0, 1, numdiv(n)*n!); \\ Michel Marcus, Jan 15 2019

a(n) = A000005(n) * n! for n > 0, a(0) = 1.

E.g.f.: 1 + Sum_{k>=1} x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 13 2019

A323307 Number of ways to fill a matrix with the parts of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 3, 12, 18, 12, 2, 36, 4, 10, 20, 72, 2, 60, 4, 40, 60, 24, 3, 120, 80, 14, 360, 120, 4, 240, 2, 240, 42, 32, 70, 720, 6, 27, 112, 480, 2, 210, 4, 84, 420, 40, 4, 1440, 280, 280, 108, 224, 5, 1260, 224, 420, 180, 22, 2, 840, 6, 72, 1680, 2880

Offset: 1

Gus Wiseman, Jan 13 2019

nonn

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The a(22) = 24 matrices:
  [111112] [111121] [111211] [112111] [121111] [211111]
.
  [111] [111] [111] [112] [121] [211]
  [112] [121] [211] [111] [111] [111]
.
  [11] [11] [11] [11] [12] [21]
  [11] [11] [12] [21] [11] [11]
  [12] [21] [11] [11] [11] [11]
.
  [1] [1] [1] [1] [1] [2]
  [1] [1] [1] [1] [2] [1]
  [1] [1] [1] [2] [1] [1]
  [1] [1] [2] [1] [1] [1]
  [1] [2] [1] [1] [1] [1]
  [2] [1] [1] [1] [1] [1]

Gus Wiseman, The a(27) = 360 matrix-arrangements of (1,1,2,2,3,3).

Cf. A007716, A056239, A112798, A120733, A181821, A305936, A318283, A318284, A323295, A323300, A323351.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Array[Length[ptnmats[Times@@Prime/@nrmptn[#]]]&,30]

a(n) = A318762(n) * A000005(A056239(n)).

A323301 Number of ways to fill a matrix with the parts of a strict integer partition of n.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 21, 25, 37, 53, 137, 153, 249, 337, 505, 845, 1085, 1497, 2061, 2785, 3661, 7589, 8849, 13329, 18033, 26017, 34225, 48773, 70805, 91977, 123765, 164761, 216373, 283205, 367913, 470889, 758793, 913825, 1264105, 1651613, 2251709, 2894793, 3927837

Offset: 0

Gus Wiseman, Jan 12 2019

nonn

			The a(6) = 21 matrices:
  [6] [1 5] [5 1] [2 4] [4 2] [1 2 3] [1 3 2] [2 1 3] [2 3 1] [3 1 2] [3 2 1]
.
  [1] [5] [2] [4]
  [5] [1] [4] [2]
.
  [1] [1] [2] [2] [3] [3]
  [2] [3] [1] [3] [1] [2]
  [3] [2] [3] [1] [2] [1]

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A000005, A000009, A000142, A053529, A294617, A321654, A323295, A323300, A323307, A323351.

b:= proc(n, i, t) option remember;
      `if`(n>i*(i+1)/2, 0, `if`(n=0, t!*numtheory[tau](t),
       b(n, i-1, t)+b(n-i, min(n-i, i-1), t+1)))
    end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 15 2019

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Table[Sum[Length[ptnmats[k]],{k,Select[Times@@Prime/@#&/@IntegerPartitions[n],SquareFreeQ]}],{n,20}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n > i(i+1)/2, 0,
     If[n == 0, t!*DivisorSigma[0, t], b[n, i - 1, t] +
     b[n - i, Min[n - i, i - 1], t + 1]]];
a[n_] := If[n == 0, 1, b[n, n, 0]];
a /@ Range[0, 50] (* Jean-François Alcover, May 13 2021, after Alois P. Heinz *)

a(n) = Sum_{y1 + ... + yk = n, y1 > ... > yk} k! * A000005(k) for n > 0, a(0) = 1.

a(0)=1 prepended by Alois P. Heinz, Jan 15 2019

A323306 Heinz numbers of integer partitions that can be arranged into a matrix with equal row-sums and equal column-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 100, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193

Offset: 1

Gus Wiseman, Jan 13 2019

nonn

First differs from A137944 in lacking 120.

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			6480 belongs to the sequence because it is the Heinz number of (3,2,2,2,2,1,1,1,1), which can be arranged in the following ways:
  [1 1 3] [1 2 2] [1 2 2] [1 3 1] [2 1 2] [2 1 2] [2 2 1] [2 2 1] [3 1 1]
  [2 2 1] [1 2 2] [3 1 1] [2 1 2] [1 3 1] [2 1 2] [1 1 3] [2 2 1] [1 2 2]
  [2 2 1] [3 1 1] [1 2 2] [2 1 2] [2 1 2] [1 3 1] [2 2 1] [1 1 3] [1 2 2]

Index entries for sequences related to Heinz numbers

Complement of A323304.
Cf. A006052, A007016, A056239, A112798, A120733, A319056, A319066, A321719.
Cf. A323300, A323302, A323347, A323348, A323349.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Select[Range[100],!Select[ptnmats[#],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]=={}&]

A323349 Number of positive integer matrices with entries summing to n, with equal row-sums and equal column-sums.

Original entry on oeis.org

1, 1, 3, 3, 6, 3, 11, 3, 12, 6, 13, 3, 52, 3, 15, 30, 57, 3, 156, 3, 238, 129, 19, 3, 2221, 6, 21, 415, 3114, 3, 14921, 3, 12853, 1044, 25, 6219, 164743, 3, 27, 2220, 851476, 3, 954088, 3, 434106, 3326714, 31, 3, 24648724, 6, 22309800, 7269, 2737618, 3, 69823653

Offset: 0

Gus Wiseman, Jan 13 2019

nonn

Also the number of non-normal semi-magic rectangles summing to n with no zeros.

Matrices must be of size m X k where m, k are divisors of n and mk <= n. This implies that a(p) = 3 for p prime, since the only allowable matrices must be of size 1 X 1, 1 X p or p X 1 with only one way to fill in the entries for each matrix size. Similarly, a(p^2) = 6 with additional allowable matrices of sizes 1 X p^2, p^2 X 1 and p X p, again with only one way to fill in the entries for each size. - Chai Wah Wu, Jan 13 2019

			The a(6) = 11 matrices:
  [6] [3 3] [2 2 2] [1 1 1 1 1 1]
.
  [3] [1 2] [2 1] [1 1 1]
  [3] [2 1] [1 2] [1 1 1]
.
  [2] [1 1]
  [2] [1 1]
  [2] [1 1]
.
  [1]
  [1]
  [1]
  [1]
  [1]
  [1]

Chai Wah Wu, Table of n, a(n) for n = 0..59

Cf. A000219, A006052, A120733, A305551, A319056, A321719, A323295, A323300, A323302, A323306, A323347, A323349.

Table[Length[Select[Join@@Table[Partition[cmp,d],{cmp,Join@@Permutations/@IntegerPartitions[n]},{d,Divisors[Length[cmp]]}],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]],{n,10}]

a(p) = 3 and a(p^2) = 6 for p prime (see comment). - Chai Wah Wu, Jan 13 2019

a(21)-a(31) from Chai Wah Wu, Jan 13 2019
a(32)-a(53) from Chai Wah Wu, Jan 14 2019
a(54) from Chai Wah Wu, Jan 16 2019

cp's OEIS Frontend

A101509 Binomial transform of tau(n) (see A000005).

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A323433 Number of ways to split an integer partition of n into consecutive subsequences of equal length.

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A323436 Number of plane partitions whose parts are the prime indices of n.

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A323351 Number of ways to fill a (not necessarily square) matrix with n zeros and ones.

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A323438 Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are weakly increasing.

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A323295 Number of ways to fill a matrix with the first n positive integers.

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A323307 Number of ways to fill a matrix with the parts of a multiset whose multiplicities are the prime indices of n.

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A323301 Number of ways to fill a matrix with the parts of a strict integer partition of n.

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