A015716 -id:A015716 - cp's OEIS frontend

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.

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

A238451 Triangle read by rows: T(n,k) is the number of k’s in all partitions of n into an even number of distinct parts.

Original entry on oeis.org

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

Offset: 1

Mircea Merca, Feb 26 2014

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

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

Columns k=1..6 are A238215, A238217, A238218, A238219, A238220, A238221.
Row sums are A238132.
Cf. A015716, A067659, A067661, A238450.

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, 0)} \\ Andrew Howroyd, Apr 29 2020

T(n,k) = Sum_{j=1..round(n/(2*k))} A067659(n-(2*j-1)*k) - Sum_{j=1..floor(n/(2*k))} A067661(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) - A238450(n,k). - Andrew Howroyd, Apr 29 2020

A325513 Heinz number of the integer partition whose parts are the multiplicities in the multiset union of all strict integer partitions of n.

Original entry on oeis.org

1, 2, 2, 8, 8, 32, 144, 432, 2160, 27000, 582120, 7623000, 336936600, 6740402760, 543454231320, 57619849046760, 4683793138766280, 412882704970215480, 88171665744392750520, 12780536107937124847320, 2685589660883755945879560, 942036670625665177379096280

Offset: 0

Gus Wiseman, May 07 2019

nonn

Also the Heinz number of row n of A015716 (with zeros removed).

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

			The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)}, with multiset union {1,1,2,2,3,4,5,6}, with multiplicities (2,2,1,1,1,1), so a(6) = prime(1)^4*prime(2)^2 = 144.
The sequence of terms together with their prime indices begins:
               1: {}
               2: {1}
               2: {1}
               8: {1,1,1}
               8: {1,1,1}
              32: {1,1,1,1,1}
             144: {1,1,1,1,2,2}
             432: {1,1,1,1,2,2,2}
            2160: {1,1,1,1,2,2,2,3}
           27000: {1,1,1,2,2,2,3,3,3}
          582120: {1,1,1,2,2,2,3,4,4,5}
         7623000: {1,1,1,2,2,3,3,3,4,5,5}
       336936600: {1,1,1,2,2,3,3,4,5,5,6,7}
      6740402760: {1,1,1,2,2,3,4,4,4,6,6,7,8}
    543454231320: {1,1,1,2,2,3,4,4,5,6,7,8,9,10}
  57619849046760: {1,1,1,2,2,3,4,5,5,6,8,9,10,11,12}

Alois P. Heinz, Table of n, a(n) for n = 0..172

Cf. A000009, A006128, A015716, A015723, A022629, A056239, A066633, A112798, A246867, A325500, A325504, A325506.

b:= proc(n, i) option remember;
      `if`(n>(i*(i+1)/2), 0, `if`(n=0, [1, 0], b(n, i-1)+
          (p-> p+[0, p[1]*x^i])(b(n-i, min(n-i, i-1)))))
    end:
a:= n-> (p-> mul((c-> `if`(c=0, 1, ithprime(c)))(
    coeff(p, x, i)), i=1..degree(p)))(b(n$2)[2]):
seq(a(n), n=0..21);  # Alois P. Heinz, Feb 23 2024

Table[Times@@Prime/@Length/@Split[Sort[Join@@Select[IntegerPartitions[n],UnsameQ@@#&]]],{n,0,15}]

a(n) = A181819(A003963(A325505(n))).

A056239(a(n)) = A015723(n).

A372888 Sum of binary ranks of all strict integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

Original entry on oeis.org

0, 1, 2, 7, 13, 31, 66, 138, 279, 581, 1173, 2375, 4783, 9630, 19316, 38802, 77689, 155673, 311639, 623845, 1248179, 2497719, 4996387, 9995304, 19992908, 39990902, 79986136, 159983241, 319975073, 639971495, 1279962115, 2559966847, 5119970499, 10240030209

Offset: 0

Gus Wiseman, May 23 2024

nonn

			The strict partitions of 6 are (6), (5,1), (4,2), (3,2,1), with respective binary ranks 32, 17, 10, 7 with sum 66, so a(6) = 66.

Alois P. Heinz, Table of n, a(n) for n = 0..3321

Row sums of A118462 (binary ranks of strict partitions).
For Heinz number the non-strict version is A145519, row sums of A215366.
For Heinz number (not binary rank) we have A147655, row sums of A246867.
The non-strict version is A372890.
A000009 counts strict partitions, ranks A005117.
A048675 gives binary rank of prime indices, distinct A087207.
A277905 groups all positive integers by binary rank of prime indices.
Binary indices (A048793):
- length A000120, complement A023416
- min A001511, opposite A000012
- max A029837 or A070939, opposite A070940
- sum A029931, product A096111
- reverse A272020
- complement A368494, sum A359400
- opposite A371572, sum A230877
- opposite complement A371571, sum A359359
Cf. A000041, A005940, A015716, A018819, A019565, A118457, A225620, A344086.

b:= proc(n, i) option remember; `if`(i*(i+1)/2 [0, p[1]*2^(i-1)]
          +p)(b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..33);  # Alois P. Heinz, May 23 2024

Table[Total[Total[2^(#-1)]& /@ Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,10}]

a(n) = Sum_{k=1..n} 2^(k-1) * A015716(n,k). - Alois P. Heinz, May 24 2024

A325515 Sum of sums of omegas of the parts over all strict integer partitions of n.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 8, 11, 14, 22, 29, 37, 50, 63, 81, 106, 129, 160, 203, 246, 303, 373, 449, 541, 654, 782, 932, 1116, 1322, 1559, 1848, 2167, 2537, 2978, 3470, 4041, 4706, 5449, 6303, 7291, 8402, 9665, 11117, 12744, 14592, 16708, 19062, 21730, 24757, 28141

Offset: 0

Gus Wiseman, May 07 2019

nonn

Also omega of the product of products of parts over all strict integer partitions of n.

The omega of n is A001222(n), the number of prime factors of n counted with multiplicity.

Cf. A001222, A003963, A015716, A015723, A022629, A056239, A066189, A112798, A147655, A246867, A325504, A325506.

Table[Sum[Total[PrimeOmega/@s],{s,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,30}]

a(n) = A001222(A325504(n)).

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A238451 Triangle read by rows: T(n,k) is the number of k’s in all partitions of n into an even number of distinct parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A325513 Heinz number of the integer partition whose parts are the multiplicities in the multiset union of all strict integer partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A372888 Sum of binary ranks of all strict integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A325515 Sum of sums of omegas of the parts over all strict integer partitions of n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula