cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A366532 Heinz numbers of integer partitions with at least one even and odd part.

Original entry on oeis.org

6, 12, 14, 15, 18, 24, 26, 28, 30, 33, 35, 36, 38, 42, 45, 48, 51, 52, 54, 56, 58, 60, 65, 66, 69, 70, 72, 74, 75, 76, 77, 78, 84, 86, 90, 93, 95, 96, 98, 99, 102, 104, 105, 106, 108, 112, 114, 116, 119, 120, 122, 123, 126, 130, 132, 135, 138, 140, 141, 142
Offset: 1

Views

Author

Gus Wiseman, Oct 16 2023

Keywords

Comments

These partitions are counted by A006477.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Examples

			The terms together with their prime indices begin:
    6: {1,2}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   42: {1,2,4}
   45: {2,2,3}
   48: {1,1,1,1,2}

Crossrefs

These partitions are counted by A006477.
Just even: A324929, counted by A047967.
Just odd: A366322, counted by A086543 (even bisection of A182616).
A031368 lists primes of odd index, even A031215.
A066207 ranks partitions with all even parts, counted by A035363.
A066208 ranks partitions with all odd parts, counted by A000009.
A112798 lists prime indices, sum A056239.
A257991 counts odd prime indices, distinct A324966.
A257992 counts even prime indices, distinct A324967.
Cf. A000720, A001222, A003963, A026804, A244991, A257992.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Or@@EvenQ/@prix[#]&&Or@@OddQ/@prix[#]&]

Formula

Intersection of A324929 and A366322.

A089004 Number of partitions of an n-element set that have at least one odd block.

Original entry on oeis.org

1, 1, 5, 11, 52, 172, 877, 3761, 21147, 109419, 678570, 4063248, 27644437, 186525861, 1382958545, 10323844183, 82864869804, 675378319788, 5832742205057, 51386368744773, 474869816156751, 4486977535640087
Offset: 1

Views

Author

Vladeta Jovovic, Nov 02 2003

Keywords

Links

Crossrefs

Cf. A000110, A005046, A003724, A086543, A087137.

Programs

  • Maple
    with(combinat):
b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
       add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1,
       max(t, `if`(j=0, 0, irem(i, 2)))), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=1..30);  # Alois P. Heinz, Mar 08 2015
  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[Cosh[x]-1](Exp[Sinh[x]]-1),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 04 2018 *)

Formula

E.g.f.: exp(cosh(x)-1)*(exp(sinh(x))-1).

A229724 Triangular array read by rows: T(n,k) is the number of partitions of n in which the greatest odd part is equal to 2k-1; n >= 1, 1 <= k <= ceiling(n/2).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 4, 2, 1, 4, 3, 1, 7, 5, 2, 1, 7, 6, 3, 1, 12, 10, 5, 2, 1, 12, 12, 7, 3, 1, 19, 18, 11, 5, 2, 1, 19, 22, 14, 7, 3, 1, 30, 31, 21, 11, 5, 2, 1, 30, 37, 27, 15, 7, 3, 1, 45, 52, 38, 22, 11, 5, 2, 1, 45, 61, 48, 29, 15, 7, 3, 1, 67, 82, 66, 41
Offset: 1

Views

Author

Geoffrey Critzer, Sep 28 2013

Keywords

Comments

Row sums are A086543.

Examples

			1;
1;
2,   1;
2,   1;
4,   2,  1;
4,   3,  1;
7,   5,  2,  1;
7,   6,  3,  1;
12, 10,  5,  2, 1;
12, 12,  7,  3, 1;
19, 18, 11,  5, 2, 1;
19, 22, 14,  7, 3, 1;
30, 31, 21, 11, 5, 2, 1;
T(7,2) = 5 because we have: 4+3 = 3+3+1 = 3+2+2 = 3+2+1+1 = 3+1+1+1+1.

Links

Crossrefs

Column k=1 gives: A025065(n-1) for n>1.

Programs

  • Maple
    b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i=1, 1+x,
       b(n, i-1) +`if`(i>n, 0, (p->`if`(irem(i, 2, 'r')=0, p,
       coeff(p, x, 0)*(1+x^(r+1)) +add(coeff(p, x, j)*x^j,
       j=r+2..degree(p))))(b(n-i, i)))))
    end:
T:= n->(p-> seq(coeff(p, x, j), j=1..degree(p)))(b(n, n)):
seq(T(n), n=1..20);  # Alois P. Heinz, Sep 28 2013
  • Mathematica
    nn=16;Map[Select[#,#>0&]&,Drop[Transpose[Table[CoefficientList[Series[x^(2k-1)/Product[1-x^j,{j,1,2k-1}] /Product[(1-x^(2j)),{j,k,nn}],{x,0,nn}],x],{k,1,nn/2}]],1]]//Grid

Formula

O.g.f. for column k: x^(2k-1)/[ prod_{j=1..2k-1}(1-x^j)*prod_{j>=k} (1-x^(2j)) ].
For even n=2j and k>=ceiling((n+2)/4) T(n,k)=A058695(j-k).
For odd n=2j-1 and k>=ceiling((n+2)/4) T(n,k)= A058696(j-k).
Previous Showing 31-33 of 33 results.

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